Calculus

Introduction to Calculus

Calculus is a section of mathematics dealing with continuous change. It encompasses several fundamental concepts: limits, derivatives, integrals, and infinite series. These ideas work together to create a powerful mathematical framework.

The core components of calculus include:
Limits - examining the behavior of functions as they approach specific values
Differential calculus - studying rates of change through derivatives
Integral calculus - analyzing accumulation and total change
Infinite series - representing functions as sums of infinite terms

Differential calculus allows us to find instantaneous rates of change and optimize functions, while integral calculus provides tools for calculating areas, volumes, and accumulated quantities. The connection between these two branches, established by the Fundamental Theorem of Calculus, creates a unified system for analyzing continuous change.

Applications of calculus extend throughout science, engineering, and economics. In physics, it models motion and energy; in engineering, it optimizes designs and processes; in economics, it analyzes rates of growth and market behavior. The subject's precise mathematical framework makes it essential for understanding and describing natural phenomena.

Calculus Formulas

The Calculus Formulas page features fundamental laws and theorems across Limits, Derivatives, Integrals, and Integration Techniques. Each entry includes step-by-step explanations, key variables, worked examples, and real-world applications - from basic limit laws and differentiation rules to advanced integration methods and improper integrals.

Two-Sided Limit Existence Theorem

\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = L \;\text{ and }\; \lim_{x \to a^+} f(x) = L

Limit of a Constant

\lim_{x \to a} c = c

Limit of the Identity Function

\lim_{x \to a} x = a

Sum and Difference Rule (Limits)

\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)

Constant Multiple Rule (Limits)

\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)

Product Rule (Limits)

\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)

Quotient Rule (Limits)

\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, \quad \lim_{x \to a} g(x) \neq 0

Power Rule (Limits)

\lim_{x \to a} [f(x)]^n = \left[\lim_{x \to a} f(x)\right]^n

Root Rule (Limits)

\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}

Absolute Value Rule (Limits)

\lim_{x \to a} |f(x)| = \left|\lim_{x \to a} f(x)\right|

Limit of a Polynomial

\lim_{x \to a} p(x) = p(a)

Limit of a Rational Function

\lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)}, \quad q(a) \neq 0

Composition Rule (Limits)

\lim_{x \to a} f(g(x)) = f\!\left(\lim_{x \to a} g(x)\right) \quad \text{if } f \text{ is continuous at } \lim_{x \to a} g(x)

Squeeze Theorem

\text{If } g(x) \leq f(x) \leq h(x) \text{ near } a \text{ and } \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L, \text{then } \lim_{x \to a} f(x) = L.

L'Hôpital's Rule

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \quad \text{for indeterminate forms } \tfrac{0}{0} \text{ or } \tfrac{\infty}{\infty}

Continuity at a Point

f \text{ is continuous at } x = a \iff \lim_{x \to a} f(x) = f(a)

One-Sided Continuity

f \text{ right-continuous at } a \iff \lim_{x \to a^+} f(x) = f(a) f \text{ left-continuous at } a \iff \lim_{x \to a^-} f(x) = f(a)

Continuity Preserved Under Operations

f, g \text{ continuous at } a \implies f \pm g, \; cf, \; f \cdot g, \; \tfrac{f}{g}\;(g(a) \neq 0), \; f \circ g \text{ continuous at } a

Intermediate Value Theorem

f \text{ continuous on } [a,b], \; k \text{ between } f(a) \text{ and } f(b) \implies \exists\, c \in (a,b) \text{ with } f(c) = k

Sine Limit at Zero

\lim_{x \to 0} \frac{\sin x}{x} = 1

Cosine Limit at Zero

\lim_{x \to 0} \frac{1 - \cos x}{x} = 0

Cosine Quadratic Limit at Zero

\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}

Tangent Limit at Zero

\lim_{x \to 0} \frac{\tan x}{x} = 1

Exponential Limit at Zero

\lim_{x \to 0} \frac{e^x - 1}{x} = 1

Logarithm Taylor Limit

\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1

Definition of e as a Limit

\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e

x ln x Limit at Zero

\lim_{x \to 0^+} x \ln x = 0

Horizontal Asymptote Condition

\lim_{x \to \infty} f(x) = L \;\text{ or }\; \lim_{x \to -\infty} f(x) = L \implies y = L \text{ is a horizontal asymptote}

Vertical Asymptote Condition

\lim_{x \to a^-} f(x) = \pm\infty \;\text{ or }\; \lim_{x \to a^+} f(x) = \pm\infty \implies x = a \text{ is a vertical asymptote}

Exponential End Behavior

\lim_{x \to \infty} e^x = \infty, \qquad \lim_{x \to -\infty} e^x = 0

Logarithm End Behavior

\lim_{x \to \infty} \ln x = \infty, \qquad \lim_{x \to 0^+} \ln x = -\infty

Exponential Beats Polynomial

\lim_{x \to \infty} \frac{x^n}{e^x} = 0 \quad \text{for any } n

Polynomial Beats Logarithm

\lim_{x \to \infty} \frac{\ln x}{x^n} = 0 \quad \text{for any } n > 0

Fundamental Theorem of Calculus, Part 1

\frac{d}{dx} \int_a^x f(t)\, dt = f(x)

Fundamental Theorem of Calculus, Part 2

\int_a^b f(x)\, dx = F(b) - F(a) \quad \text{where } F'(x) = f(x)

Antiderivative Family

\int f(x)\, dx = F(x) + C \quad \text{where } F'(x) = f(x)

Sum and Difference Rule (Integrals)

\int [f(x) \pm g(x)]\, dx = \int f(x)\, dx \pm \int g(x)\, dx

Constant Multiple Rule (Integrals)

\int c \cdot f(x)\, dx = c \int f(x)\, dx

Additivity Over Intervals

\int_a^b f(x)\, dx + \int_b^c f(x)\, dx = \int_a^c f(x)\, dx

Reversing Limits of Integration

\int_a^b f(x)\, dx = -\int_b^a f(x)\, dx

Zero-Width Interval

\int_a^a f(x)\, dx = 0

Comparison Property (Integrals)

f(x) \leq g(x) \text{ on } [a, b] \implies \int_a^b f(x)\, dx \leq \int_a^b g(x)\, dx

Bounding Property (Integrals)

m \leq f(x) \leq M \text{ on } [a, b] \implies m(b - a) \leq \int_a^b f(x)\, dx \leq M(b - a)

Substitution Rule

\int f(g(x))\, g'(x)\, dx = \int f(u)\, du \quad \text{where } u = g(x)

Integration by Parts

\int u\, dv = uv - \int v\, du

Total Unsigned Area

\text{Total area} = \int_a^b |f(x)|\, dx

Power Rule (Integrals)

\int x^n\, dx = \frac{x^{n+1}}{n + 1} + C \quad (n \neq -1)

Reciprocal Antiderivative

\int \frac{1}{x}\, dx = \ln|x| + C

Logarithmic Derivative Pattern

\int \frac{f'(x)}{f(x)}\, dx = \ln|f(x)| + C

Antiderivative of Natural Log

\int \ln x\, dx = x \ln x - x + C

Exponential Antiderivative

\int e^x\, dx = e^x + C

General Exponential Antiderivative

\int a^x\, dx = \frac{a^x}{\ln a} + C \quad (a > 0,\, a \neq 1)

Antiderivative of Sine

\int \sin x\, dx = -\cos x + C

Antiderivative of Cosine

\int \cos x\, dx = \sin x + C

Antiderivative of Secant Squared

\int \sec^2 x\, dx = \tan x + C

Antiderivative of Cosecant Squared

\int \csc^2 x\, dx = -\cot x + C

Antiderivative of Sec Tan

\int \sec x \tan x\, dx = \sec x + C

Antiderivative of Csc Cot

\int \csc x \cot x\, dx = -\csc x + C

Antiderivative of Tangent

\int \tan x\, dx = \ln|\sec x| + C

Antiderivative of Cotangent

\int \cot x\, dx = \ln|\sin x| + C

Antiderivative of Secant

\int \sec x\, dx = \ln|\sec x + \tan x| + C

Antiderivative of Cosecant

\int \csc x\, dx = \ln|\csc x - \cot x| + C

Arctangent Form

\int \frac{1}{a^2 + x^2}\, dx = \frac{1}{a} \arctan\frac{x}{a} + C

Arcsine Form

\int \frac{1}{\sqrt{a^2 - x^2}}\, dx = \arcsin\frac{x}{a} + C

Arcsecant Form

\int \frac{1}{x \sqrt{x^2 - 1}}\, dx = \text{arcsec}\,|x| + C

Even Function Symmetry

\int_{-a}^{a} f(x)\, dx = 2 \int_0^a f(x)\, dx \quad \text{if } f \text{ is even}

Odd Function Symmetry

\int_{-a}^{a} f(x)\, dx = 0 \quad \text{if } f \text{ is odd}

Improper Integral (Infinite Limits)

\int_a^{\infty} f(x)\, dx = \lim_{b \to \infty} \int_a^b f(x)\, dx

Improper Integral (Discontinuous Integrand)

\int_a^b f(x)\, dx = \lim_{t \to b^-} \int_a^t f(x)\, dx \quad \text{(asymptote at } b\text{)}

p-Test for Improper Integrals

\int_1^{\infty} \frac{1}{x^p}\, dx \;\begin{cases} \text{converges} & p > 1 \\ \text{diverges} & p \leq 1 \end{cases} \qquad \int_0^1 \frac{1}{x^p}\, dx \;\begin{cases} \text{converges} & p < 1 \\ \text{diverges} & p \geq 1 \end{cases}

Average Value of a Function

f_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x)\, dx

Average Rate of Change

\bar{m} = \frac{f(b) - f(a)}{b - a}

Derivative Limit Definition

f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim_{t \to x} \frac{f(t) - f(x)}{t - x}

Constant Rule (Derivatives)

\frac{d}{dx}[c] = 0

Power Rule (Derivatives)

\frac{d}{dx}[x^n] = n x^{n-1}

Constant Multiple Rule (Derivatives)

\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)

Sum and Difference Rule (Derivatives)

\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)

Product Rule (Derivatives)

\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)

Quotient Rule (Derivatives)

\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}

Chain Rule

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \qquad \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

Mean Value Theorem

f'(c) = \frac{f(b) - f(a)}{b - a} \qquad \text{for some } c \in (a, b)

Rolle's Theorem

\text{If } f(a) = f(b), \text{ then } f'(c) = 0 \text{ for some } c \in (a, b)

Derivative of Sine

\frac{d}{dx}[\sin x] = \cos x

Derivative of Cosine

\frac{d}{dx}[\cos x] = -\sin x

Derivative of Tangent

\frac{d}{dx}[\tan x] = \sec^2 x

Derivative of Cotangent

\frac{d}{dx}[\cot x] = -\csc^2 x

Derivative of Secant

\frac{d}{dx}[\sec x] = \sec x \tan x

Derivative of Cosecant

\frac{d}{dx}[\csc x] = -\csc x \cot x

Derivative of Natural Exponential

\frac{d}{dx}[e^x] = e^x

Derivative of General Exponential

\frac{d}{dx}[a^x] = a^x \ln a

Derivative of Natural Logarithm

\frac{d}{dx}[\ln x] = \frac{1}{x}

Derivative of General Logarithm

\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}

Derivative of Arcsine

\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1 - x^2}}

Derivative of Arccosine

\frac{d}{dx}[\arccos x] = -\frac{1}{\sqrt{1 - x^2}}

Derivative of Arctangent

\frac{d}{dx}[\arctan x] = \frac{1}{1 + x^2}

Derivative of Arccotangent

\frac{d}{dx}[\operatorname{arccot} x] = -\frac{1}{1 + x^2}

Derivative of Arcsecant

\frac{d}{dx}[\operatorname{arcsec} x] = \frac{1}{|x| \sqrt{x^2 - 1}}

Derivative of Arccosecant

\frac{d}{dx}[\operatorname{arccsc} x] = -\frac{1}{|x| \sqrt{x^2 - 1}}

Derivative of Hyperbolic Sine

\frac{d}{dx}[\sinh x] = \cosh x

Derivative of Hyperbolic Cosine

\frac{d}{dx}[\cosh x] = \sinh x

Derivative of Hyperbolic Tangent

\frac{d}{dx}[\tanh x] = \operatorname{sech}^2 x

Derivative of Hyperbolic Cotangent

\frac{d}{dx}[\coth x] = -\operatorname{csch}^2 x

Derivative of Hyperbolic Secant

\frac{d}{dx}[\operatorname{sech} x] = -\operatorname{sech} x \tanh x

Derivative of Hyperbolic Cosecant

\frac{d}{dx}[\operatorname{csch} x] = -\operatorname{csch} x \coth x

Derivative of Inverse Hyperbolic Sine

\frac{d}{dx}[\operatorname{arcsinh} x] = \frac{1}{\sqrt{x^2 + 1}}

Derivative of Inverse Hyperbolic Cosine

\frac{d}{dx}[\operatorname{arccosh} x] = \frac{1}{\sqrt{x^2 - 1}}

Derivative of Inverse Hyperbolic Tangent

\frac{d}{dx}[\operatorname{arctanh} x] = \frac{1}{1 - x^2}

One-Sided Derivative

f'_-(a) = \lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h} \qquad f'_+(a) = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}

Differentiability Implies Continuity

f \text{ differentiable at } a \implies f \text{ continuous at } a

Differential

dy = f'(x)\, dx

Linear Approximation

f(x) \approx f(a) + f'(a)(x - a) \qquad \Delta y \approx f'(a)\, \Delta x

Total Differential

dz = \frac{\partial z}{\partial x}\, dx + \frac{\partial z}{\partial y}\, dy

Logarithmic Derivative

\frac{d}{dx}[\ln f(x)] = \frac{f'(x)}{f(x)}

Tangent Line Equation

y - f(a) = f'(a)(x - a) \qquad y = f(a) + f'(a)(x - a)

Normal Line Equation

y - f(a) = -\frac{1}{f'(a)}(x - a)

Sign of First Derivative

f'(x) > 0 \implies f \text{ increasing}, \quad f'(x) < 0 \implies f \text{ decreasing}

Critical Point Condition

f'(c) = 0 \quad \text{or} \quad f'(c) \text{ undefined}

First Derivative Test

f'(x) \text{ changes } + \to - \text{ at } c \implies c \text{ is a local max} f'(x) \text{ changes } - \to + \text{ at } c \implies c \text{ is a local min}

Second Derivative Test

f'(c) = 0, \; f''(c) > 0 \implies c \text{ is a local min} f'(c) = 0, \; f''(c) < 0 \implies c \text{ is a local max}

Concavity from Second Derivative

f''(x) > 0 \implies f \text{ concave up}, \quad f''(x) < 0 \implies f \text{ concave down}

Inflection Point Condition

f''(c) = 0 \text{ or undefined}, \text{ and } f''(x) \text{ changes sign at } c

Extreme Value Theorem

f \text{ continuous on } [a, b] \implies f \text{ attains a max and min on } [a, b]

nth Derivative of Power

\frac{d^n}{dx^n}[x^m] = \frac{m!}{(m-n)!}\, x^{m-n} \quad (n \leq m)

nth Derivative of Natural Exponential

\frac{d^n}{dx^n}[e^x] = e^x

nth Derivative of Scaled Exponential

\frac{d^n}{dx^n}[e^{ax}] = a^n e^{ax}

nth Derivative of Sine

\frac{d^n}{dx^n}[\sin x] = \sin\!\left(x + \frac{n\pi}{2}\right)

nth Derivative of Cosine

\frac{d^n}{dx^n}[\cos x] = \cos\!\left(x + \frac{n\pi}{2}\right)

nth Derivative of Reciprocal

\frac{d^n}{dx^n}\!\left[\frac{1}{x}\right] = \frac{(-1)^n\, n!}{x^{n+1}}

nth Derivative of Natural Logarithm

\frac{d^n}{dx^n}[\ln x] = \frac{(-1)^{n-1}\, (n-1)!}{x^n}

Taylor Series

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n

Inverse Function Derivative

(f^{-1})'(b) = \frac{1}{f'(a)} \quad \text{where } b = f(a)

Logarithmic Differentiation

y = f(x) \implies \ln y = \ln f(x) \implies \frac{y'}{y} = \frac{d}{dx}[\ln f(x)] \implies y' = y \cdot \frac{d}{dx}[\ln f(x)]

Parametric First Derivative

\frac{dy}{dx} = \frac{dy/dt}{dx/dt} \quad \text{when } x = x(t), \, y = y(t)

Parametric Second Derivative

\frac{d^2 y}{dx^2} = \frac{d}{dx}\!\left[\frac{dy}{dx}\right] = \frac{d/dt\,[dy/dx]}{dx/dt}

Two-Sided Limit Existence Theorem

\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = L \;\text{ and }\; \lim_{x \to a^+} f(x) = L

Limit of a Constant

\lim_{x \to a} c = c

Limit of the Identity Function

\lim_{x \to a} x = a

Sum and Difference Rule (Limits)

\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)

Constant Multiple Rule (Limits)

\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)

Product Rule (Limits)

\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)

Quotient Rule (Limits)

\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, \quad \lim_{x \to a} g(x) \neq 0

Power Rule (Limits)

\lim_{x \to a} [f(x)]^n = \left[\lim_{x \to a} f(x)\right]^n

Root Rule (Limits)

\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}

Absolute Value Rule (Limits)

\lim_{x \to a} |f(x)| = \left|\lim_{x \to a} f(x)\right|

Limit of a Polynomial

\lim_{x \to a} p(x) = p(a)

Limit of a Rational Function

\lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)}, \quad q(a) \neq 0

Composition Rule (Limits)

\lim_{x \to a} f(g(x)) = f\!\left(\lim_{x \to a} g(x)\right) \quad \text{if } f \text{ is continuous at } \lim_{x \to a} g(x)

Squeeze Theorem

\text{If } g(x) \leq f(x) \leq h(x) \text{ near } a \text{ and } \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L, \text{then } \lim_{x \to a} f(x) = L.

L'Hôpital's Rule

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \quad \text{for indeterminate forms } \tfrac{0}{0} \text{ or } \tfrac{\infty}{\infty}

Continuity at a Point

f \text{ is continuous at } x = a \iff \lim_{x \to a} f(x) = f(a)

One-Sided Continuity

f \text{ right-continuous at } a \iff \lim_{x \to a^+} f(x) = f(a) f \text{ left-continuous at } a \iff \lim_{x \to a^-} f(x) = f(a)

Continuity Preserved Under Operations

f, g \text{ continuous at } a \implies f \pm g, \; cf, \; f \cdot g, \; \tfrac{f}{g}\;(g(a) \neq 0), \; f \circ g \text{ continuous at } a

Intermediate Value Theorem

f \text{ continuous on } [a,b], \; k \text{ between } f(a) \text{ and } f(b) \implies \exists\, c \in (a,b) \text{ with } f(c) = k

Sine Limit at Zero

\lim_{x \to 0} \frac{\sin x}{x} = 1

Cosine Limit at Zero

\lim_{x \to 0} \frac{1 - \cos x}{x} = 0

Cosine Quadratic Limit at Zero

\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}

Tangent Limit at Zero

\lim_{x \to 0} \frac{\tan x}{x} = 1

Exponential Limit at Zero

\lim_{x \to 0} \frac{e^x - 1}{x} = 1

Logarithm Taylor Limit

\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1

Definition of e as a Limit

\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e

x ln x Limit at Zero

\lim_{x \to 0^+} x \ln x = 0

Horizontal Asymptote Condition

\lim_{x \to \infty} f(x) = L \;\text{ or }\; \lim_{x \to -\infty} f(x) = L \implies y = L \text{ is a horizontal asymptote}

Vertical Asymptote Condition

\lim_{x \to a^-} f(x) = \pm\infty \;\text{ or }\; \lim_{x \to a^+} f(x) = \pm\infty \implies x = a \text{ is a vertical asymptote}

Exponential End Behavior

\lim_{x \to \infty} e^x = \infty, \qquad \lim_{x \to -\infty} e^x = 0

Logarithm End Behavior

\lim_{x \to \infty} \ln x = \infty, \qquad \lim_{x \to 0^+} \ln x = -\infty

Exponential Beats Polynomial

\lim_{x \to \infty} \frac{x^n}{e^x} = 0 \quad \text{for any } n

Polynomial Beats Logarithm

\lim_{x \to \infty} \frac{\ln x}{x^n} = 0 \quad \text{for any } n > 0

Fundamental Theorem of Calculus, Part 1

\frac{d}{dx} \int_a^x f(t)\, dt = f(x)

Fundamental Theorem of Calculus, Part 2

\int_a^b f(x)\, dx = F(b) - F(a) \quad \text{where } F'(x) = f(x)

Antiderivative Family

\int f(x)\, dx = F(x) + C \quad \text{where } F'(x) = f(x)

Sum and Difference Rule (Integrals)

\int [f(x) \pm g(x)]\, dx = \int f(x)\, dx \pm \int g(x)\, dx

Constant Multiple Rule (Integrals)

\int c \cdot f(x)\, dx = c \int f(x)\, dx

Additivity Over Intervals

\int_a^b f(x)\, dx + \int_b^c f(x)\, dx = \int_a^c f(x)\, dx

Reversing Limits of Integration

\int_a^b f(x)\, dx = -\int_b^a f(x)\, dx

Zero-Width Interval

\int_a^a f(x)\, dx = 0

Comparison Property (Integrals)

f(x) \leq g(x) \text{ on } [a, b] \implies \int_a^b f(x)\, dx \leq \int_a^b g(x)\, dx

Bounding Property (Integrals)

m \leq f(x) \leq M \text{ on } [a, b] \implies m(b - a) \leq \int_a^b f(x)\, dx \leq M(b - a)

Substitution Rule

\int f(g(x))\, g'(x)\, dx = \int f(u)\, du \quad \text{where } u = g(x)

Integration by Parts

\int u\, dv = uv - \int v\, du

Total Unsigned Area

\text{Total area} = \int_a^b |f(x)|\, dx

Power Rule (Integrals)

\int x^n\, dx = \frac{x^{n+1}}{n + 1} + C \quad (n \neq -1)

Reciprocal Antiderivative

\int \frac{1}{x}\, dx = \ln|x| + C

Logarithmic Derivative Pattern

\int \frac{f'(x)}{f(x)}\, dx = \ln|f(x)| + C

Antiderivative of Natural Log

\int \ln x\, dx = x \ln x - x + C

Exponential Antiderivative

\int e^x\, dx = e^x + C

General Exponential Antiderivative

\int a^x\, dx = \frac{a^x}{\ln a} + C \quad (a > 0,\, a \neq 1)

Antiderivative of Sine

\int \sin x\, dx = -\cos x + C

Antiderivative of Cosine

\int \cos x\, dx = \sin x + C

Antiderivative of Secant Squared

\int \sec^2 x\, dx = \tan x + C

Antiderivative of Cosecant Squared

\int \csc^2 x\, dx = -\cot x + C

Antiderivative of Sec Tan

\int \sec x \tan x\, dx = \sec x + C

Antiderivative of Csc Cot

\int \csc x \cot x\, dx = -\csc x + C

Antiderivative of Tangent

\int \tan x\, dx = \ln|\sec x| + C

Antiderivative of Cotangent

\int \cot x\, dx = \ln|\sin x| + C

Antiderivative of Secant

\int \sec x\, dx = \ln|\sec x + \tan x| + C

Antiderivative of Cosecant

\int \csc x\, dx = \ln|\csc x - \cot x| + C

Arctangent Form

\int \frac{1}{a^2 + x^2}\, dx = \frac{1}{a} \arctan\frac{x}{a} + C

Arcsine Form

\int \frac{1}{\sqrt{a^2 - x^2}}\, dx = \arcsin\frac{x}{a} + C

Arcsecant Form

\int \frac{1}{x \sqrt{x^2 - 1}}\, dx = \text{arcsec}\,|x| + C

Even Function Symmetry

\int_{-a}^{a} f(x)\, dx = 2 \int_0^a f(x)\, dx \quad \text{if } f \text{ is even}

Odd Function Symmetry

\int_{-a}^{a} f(x)\, dx = 0 \quad \text{if } f \text{ is odd}

Improper Integral (Infinite Limits)

\int_a^{\infty} f(x)\, dx = \lim_{b \to \infty} \int_a^b f(x)\, dx

Improper Integral (Discontinuous Integrand)

\int_a^b f(x)\, dx = \lim_{t \to b^-} \int_a^t f(x)\, dx \quad \text{(asymptote at } b\text{)}

p-Test for Improper Integrals

\int_1^{\infty} \frac{1}{x^p}\, dx \;\begin{cases} \text{converges} & p > 1 \\ \text{diverges} & p \leq 1 \end{cases} \qquad \int_0^1 \frac{1}{x^p}\, dx \;\begin{cases} \text{converges} & p < 1 \\ \text{diverges} & p \geq 1 \end{cases}

Average Value of a Function

f_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x)\, dx

Average Rate of Change

\bar{m} = \frac{f(b) - f(a)}{b - a}

Derivative Limit Definition

f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim_{t \to x} \frac{f(t) - f(x)}{t - x}

Constant Rule (Derivatives)

\frac{d}{dx}[c] = 0

Power Rule (Derivatives)

\frac{d}{dx}[x^n] = n x^{n-1}

Constant Multiple Rule (Derivatives)

\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)

Sum and Difference Rule (Derivatives)

\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)

Product Rule (Derivatives)

\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)

Quotient Rule (Derivatives)

\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}

Chain Rule

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \qquad \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

Mean Value Theorem

f'(c) = \frac{f(b) - f(a)}{b - a} \qquad \text{for some } c \in (a, b)

Rolle's Theorem

\text{If } f(a) = f(b), \text{ then } f'(c) = 0 \text{ for some } c \in (a, b)

Derivative of Sine

\frac{d}{dx}[\sin x] = \cos x

Derivative of Cosine

\frac{d}{dx}[\cos x] = -\sin x

Derivative of Tangent

\frac{d}{dx}[\tan x] = \sec^2 x

Derivative of Cotangent

\frac{d}{dx}[\cot x] = -\csc^2 x

Derivative of Secant

\frac{d}{dx}[\sec x] = \sec x \tan x

Derivative of Cosecant

\frac{d}{dx}[\csc x] = -\csc x \cot x

Derivative of Natural Exponential

\frac{d}{dx}[e^x] = e^x

Derivative of General Exponential

\frac{d}{dx}[a^x] = a^x \ln a

Derivative of Natural Logarithm

\frac{d}{dx}[\ln x] = \frac{1}{x}

Derivative of General Logarithm

\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}

Derivative of Arcsine

\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1 - x^2}}

Derivative of Arccosine

\frac{d}{dx}[\arccos x] = -\frac{1}{\sqrt{1 - x^2}}

Derivative of Arctangent

\frac{d}{dx}[\arctan x] = \frac{1}{1 + x^2}

Derivative of Arccotangent

\frac{d}{dx}[\operatorname{arccot} x] = -\frac{1}{1 + x^2}

Derivative of Arcsecant

\frac{d}{dx}[\operatorname{arcsec} x] = \frac{1}{|x| \sqrt{x^2 - 1}}

Derivative of Arccosecant

\frac{d}{dx}[\operatorname{arccsc} x] = -\frac{1}{|x| \sqrt{x^2 - 1}}

Derivative of Hyperbolic Sine

\frac{d}{dx}[\sinh x] = \cosh x

Derivative of Hyperbolic Cosine

\frac{d}{dx}[\cosh x] = \sinh x

Derivative of Hyperbolic Tangent

\frac{d}{dx}[\tanh x] = \operatorname{sech}^2 x

Derivative of Hyperbolic Cotangent

\frac{d}{dx}[\coth x] = -\operatorname{csch}^2 x

Derivative of Hyperbolic Secant

\frac{d}{dx}[\operatorname{sech} x] = -\operatorname{sech} x \tanh x

Derivative of Hyperbolic Cosecant

\frac{d}{dx}[\operatorname{csch} x] = -\operatorname{csch} x \coth x

Derivative of Inverse Hyperbolic Sine

\frac{d}{dx}[\operatorname{arcsinh} x] = \frac{1}{\sqrt{x^2 + 1}}

Derivative of Inverse Hyperbolic Cosine

\frac{d}{dx}[\operatorname{arccosh} x] = \frac{1}{\sqrt{x^2 - 1}}

Derivative of Inverse Hyperbolic Tangent

\frac{d}{dx}[\operatorname{arctanh} x] = \frac{1}{1 - x^2}

One-Sided Derivative

f'_-(a) = \lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h} \qquad f'_+(a) = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}

Differentiability Implies Continuity

f \text{ differentiable at } a \implies f \text{ continuous at } a

Differential

dy = f'(x)\, dx

Linear Approximation

f(x) \approx f(a) + f'(a)(x - a) \qquad \Delta y \approx f'(a)\, \Delta x

Total Differential

dz = \frac{\partial z}{\partial x}\, dx + \frac{\partial z}{\partial y}\, dy

Logarithmic Derivative

\frac{d}{dx}[\ln f(x)] = \frac{f'(x)}{f(x)}

Tangent Line Equation

y - f(a) = f'(a)(x - a) \qquad y = f(a) + f'(a)(x - a)

Normal Line Equation

y - f(a) = -\frac{1}{f'(a)}(x - a)

Sign of First Derivative

f'(x) > 0 \implies f \text{ increasing}, \quad f'(x) < 0 \implies f \text{ decreasing}

Critical Point Condition

f'(c) = 0 \quad \text{or} \quad f'(c) \text{ undefined}

First Derivative Test

f'(x) \text{ changes } + \to - \text{ at } c \implies c \text{ is a local max} f'(x) \text{ changes } - \to + \text{ at } c \implies c \text{ is a local min}

Second Derivative Test

f'(c) = 0, \; f''(c) > 0 \implies c \text{ is a local min} f'(c) = 0, \; f''(c) < 0 \implies c \text{ is a local max}

Concavity from Second Derivative

f''(x) > 0 \implies f \text{ concave up}, \quad f''(x) < 0 \implies f \text{ concave down}

Inflection Point Condition

f''(c) = 0 \text{ or undefined}, \text{ and } f''(x) \text{ changes sign at } c

Extreme Value Theorem

f \text{ continuous on } [a, b] \implies f \text{ attains a max and min on } [a, b]

nth Derivative of Power

\frac{d^n}{dx^n}[x^m] = \frac{m!}{(m-n)!}\, x^{m-n} \quad (n \leq m)

nth Derivative of Natural Exponential

\frac{d^n}{dx^n}[e^x] = e^x

nth Derivative of Scaled Exponential

\frac{d^n}{dx^n}[e^{ax}] = a^n e^{ax}

nth Derivative of Sine

\frac{d^n}{dx^n}[\sin x] = \sin\!\left(x + \frac{n\pi}{2}\right)

nth Derivative of Cosine

\frac{d^n}{dx^n}[\cos x] = \cos\!\left(x + \frac{n\pi}{2}\right)

nth Derivative of Reciprocal

\frac{d^n}{dx^n}\!\left[\frac{1}{x}\right] = \frac{(-1)^n\, n!}{x^{n+1}}

nth Derivative of Natural Logarithm

\frac{d^n}{dx^n}[\ln x] = \frac{(-1)^{n-1}\, (n-1)!}{x^n}

Taylor Series

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n

Inverse Function Derivative

(f^{-1})'(b) = \frac{1}{f'(a)} \quad \text{where } b = f(a)

Logarithmic Differentiation

y = f(x) \implies \ln y = \ln f(x) \implies \frac{y'}{y} = \frac{d}{dx}[\ln f(x)] \implies y' = y \cdot \frac{d}{dx}[\ln f(x)]

Parametric First Derivative

\frac{dy}{dx} = \frac{dy/dt}{dx/dt} \quad \text{when } x = x(t), \, y = y(t)

Parametric Second Derivative

\frac{d^2 y}{dx^2} = \frac{d}{dx}\!\left[\frac{dy}{dx}\right] = \frac{d/dt\,[dy/dx]}{dx/dt}

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Calculus Terms and Definitions

Limit

The value that $f(x)$ approaches as $x$ approaches a specified point $a$: $\lim_{x \to a} f(x) = L$ means $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$.

One-Sided Limit

The value $f(x)$ approaches as $x$ approaches $a$ from one direction only: $\lim_{x \to a^-} f(x)$ (from the left) or $\lim_{x \to a^+} f(x)$ (from the right).

Continuity

A function $f$ is continuous at $x = a$ if three conditions hold: $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and $\lim_{x \to a} f(x) = f(a)$.

Discontinuity

A point where a function fails to be continuous — at least one of the three continuity conditions is violated.

Indeterminate Form

An expression arising from direct substitution in a limit whose value cannot be determined without further analysis: $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $\infty - \infty$, $0^0$, $1^\infty$, $\infty^0$.

Asymptote

A line that the graph of a function approaches arbitrarily closely as $x$ or $f(x)$ tends toward infinity or a boundary point.

Derivative

The instantaneous rate of change of $f$ at $x = a$, defined as $f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$, when this limit exists.

Differentiability

A function $f$ is differentiable at $x = a$ if $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ exists and is finite.

Differential

The independent differential $dx$ is a freely chosen increment in $x$; the dependent differential is $dy = f'(x) \cdot dx$, the change predicted by the tangent line.

Higher-Order Derivative

The $n$th derivative $f^{(n)}(x)$, obtained by differentiating $f$ a total of $n$ times: $f''(x) = \frac{d^2y}{dx^2}$, $f'''(x) = \frac{d^3y}{dx^3}$, and so on.

Partial Derivative

The derivative of a multivariable function with respect to one variable while all others are held constant: $\frac{\partial f}{\partial x}$.

Instantaneous Rate of Change

The rate of change of $f$ at a single point $x = a$, equal to the derivative $f'(a)$: the limit of average rates of change as the interval shrinks to zero.

Average Rate of Change

The ratio $\frac{f(b) - f(a)}{b - a}$, measuring the overall change in $f$ per unit change in input over the interval $[a, b]$.

Tangent Line

The line through $(a, f(a))$ with slope $f'(a)$: $y - f(a) = f'(a)(x - a)$.

Critical Point

A value $x = c$ in the domain of $f$ where $f'(c) = 0$ or $f'(c)$ does not exist.

Local Extremum

A point where $f$ achieves a value greater than (local maximum) or less than (local minimum) all nearby values: $f(c) \geq f(x)$ or $f(c) \leq f(x)$ for all $x$ in some open interval around $c$.

Concavity

A property describing how the slope of $f$ changes: concave up where $f''(x) > 0$ (slope increasing), concave down where $f''(x) < 0$ (slope decreasing).

Inflection Point

A point on the graph of $f$ where the concavity changes — from concave up to concave down, or the reverse.

Monotonic Function

A function that is entirely non-decreasing or entirely non-increasing on an interval. Strictly monotonic: strictly increasing ($a < b \implies f(a) < f(b)$) or strictly decreasing ($a < b \implies f(a) > f(b)$).

Antiderivative

A function $F$ whose derivative equals the given function: $F'(x) = f(x)$. Also called a primitive.

Indefinite Integral

The general antiderivative of $f$, written $\int f(x)\,dx = F(x) + C$, representing the entire family of functions whose derivative is $f$.

Definite Integral

The limit of Riemann sums over $[a, b]$: $\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x$, yielding a number that represents accumulated signed area.

Integrand

The function $f(x)$ appearing inside an integral expression $\int f(x)\,dx$ — the function being integrated.

Bounds of Integration

The values $a$ (lower bound) and $b$ (upper bound) in a definite integral $\int_a^b f(x)\,dx$, specifying where accumulation begins and ends.

Riemann Sum

An approximation to the definite integral formed by partitioning $[a, b]$ into subintervals and summing rectangular areas: $S_n = \sum_{i=1}^{n} f(x_i^*) \Delta x$.

Improper Integral

A definite integral where the interval is infinite or the integrand is unbounded within the interval, evaluated as a limit of proper integrals.

Signed Area

The value of a definite integral interpreted geometrically: area above the $x$-axis counts as positive, area below counts as negative.

Average Value of a Function

The mean output of $f$ over $[a, b]$: $f_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x)\,dx$.

Limit

The value that $f(x)$ approaches as $x$ approaches a specified point $a$: $\lim_{x \to a} f(x) = L$ means $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$.

One-Sided Limit

The value $f(x)$ approaches as $x$ approaches $a$ from one direction only: $\lim_{x \to a^-} f(x)$ (from the left) or $\lim_{x \to a^+} f(x)$ (from the right).

Continuity

A function $f$ is continuous at $x = a$ if three conditions hold: $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and $\lim_{x \to a} f(x) = f(a)$.

Discontinuity

A point where a function fails to be continuous — at least one of the three continuity conditions is violated.

Indeterminate Form

Asymptote

A line that the graph of a function approaches arbitrarily closely as $x$ or $f(x)$ tends toward infinity or a boundary point.

Derivative

The instantaneous rate of change of $f$ at $x = a$, defined as $f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$, when this limit exists.

Differentiability

A function $f$ is differentiable at $x = a$ if $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ exists and is finite.

Differential

The independent differential $dx$ is a freely chosen increment in $x$; the dependent differential is $dy = f'(x) \cdot dx$, the change predicted by the tangent line.

Higher-Order Derivative

The $n$th derivative $f^{(n)}(x)$, obtained by differentiating $f$ a total of $n$ times: $f''(x) = \frac{d^2y}{dx^2}$, $f'''(x) = \frac{d^3y}{dx^3}$, and so on.

Partial Derivative

The derivative of a multivariable function with respect to one variable while all others are held constant: $\frac{\partial f}{\partial x}$.

Instantaneous Rate of Change

The rate of change of $f$ at a single point $x = a$, equal to the derivative $f'(a)$: the limit of average rates of change as the interval shrinks to zero.

Average Rate of Change

The ratio $\frac{f(b) - f(a)}{b - a}$, measuring the overall change in $f$ per unit change in input over the interval $[a, b]$.

Tangent Line

The line through $(a, f(a))$ with slope $f'(a)$: $y - f(a) = f'(a)(x - a)$.

Critical Point

A value $x = c$ in the domain of $f$ where $f'(c) = 0$ or $f'(c)$ does not exist.

Local Extremum

A point where $f$ achieves a value greater than (local maximum) or less than (local minimum) all nearby values: $f(c) \geq f(x)$ or $f(c) \leq f(x)$ for all $x$ in some open interval around $c$.

Concavity

A property describing how the slope of $f$ changes: concave up where $f''(x) > 0$ (slope increasing), concave down where $f''(x) < 0$ (slope decreasing).

Inflection Point

A point on the graph of $f$ where the concavity changes — from concave up to concave down, or the reverse.

Monotonic Function

Antiderivative

A function $F$ whose derivative equals the given function: $F'(x) = f(x)$. Also called a primitive.

Indefinite Integral

The general antiderivative of $f$, written $\int f(x)\,dx = F(x) + C$, representing the entire family of functions whose derivative is $f$.

Definite Integral

The limit of Riemann sums over $[a, b]$: $\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x$, yielding a number that represents accumulated signed area.

Integrand

The function $f(x)$ appearing inside an integral expression $\int f(x)\,dx$ — the function being integrated.

Bounds of Integration

The values $a$ (lower bound) and $b$ (upper bound) in a definite integral $\int_a^b f(x)\,dx$, specifying where accumulation begins and ends.

Riemann Sum

An approximation to the definite integral formed by partitioning $[a, b]$ into subintervals and summing rectangular areas: $S_n = \sum_{i=1}^{n} f(x_i^*) \Delta x$.

Improper Integral

A definite integral where the interval is infinite or the integrand is unbounded within the interval, evaluated as a limit of proper integrals.

Signed Area

The value of a definite integral interpreted geometrically: area above the $x$-axis counts as positive, area below counts as negative.

Average Value of a Function

The mean output of $f$ over $[a, b]$: $f_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x)\,dx$.

See All Definitions

The Calculus Terms and Definitions page provides a comprehensive collection of essential calculus concepts organized across multiple categories including Functions, Differentiation, Integration, Geometry, Motion and Dynamics, and Vector Calculus. From fundamental concepts like derivatives and integrals to advanced topics in vector analysis and differential equations, each term is clearly defined to support understanding of calculus principles and their applications.

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Limits, Derivatives, and Integrals

Limits play a central role in defining both derivatives and integrals.They provide the precise language for handling values that are approached but not necessarily reached.
A derivative is defined as the limit of the average rate of change while the interval shrinks to zero and, in such a way, indicates how function changes at a specific point.
An integral, on the other hand, measures accumulation—such as area under a curve or total distance traveled. The concept of a limit is also used here, serving as the key tool for defining the integral through an infinite sum of infinitesimally small quantities.
What ties them all together is the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse processes: taking the derivative of an integral function returns the original function, and integrating a derivative recovers accumulated change.
In essence, limits allow us to rigorously define both change (via derivatives) and accumulation (via integrals), revealing a deep unity among these core ideas of calculus.

Calculus Symbols Reference

Our Calculus Symbols page offers a detailed catalog of notation used in differential and integral calculus. This comprehensive resource organizes symbols by their mathematical functions to help students and professionals navigate the language of calculus.
The reference covers essential notation across categories including differentiation (f'(x), df/dx, ∇f), integration (∫, ∬, ∮), limits (limₓ→c), and infinite series (∑). Advanced topics include vector calculus notation for divergence and curl, differential operators like the Laplacian (∇²f), and specialized notation for curvature and differential equations.
Each symbol is presented with its proper LaTeX representation and a concise explanation of its meaning, making this an essential resource for anyone working with calculus concepts in academic or professional settings.

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