What is the derivative of a constant?

The derivative of any constant c is zero: d/dx[c] = 0. A constant function graphs as a horizontal line with slope zero everywhere. Constant terms vanish under differentiation.

What is the power rule for derivatives?

For any real exponent n: d/dx[xⁿ] = nxⁿ⁻¹. The rule works for positive integers, negative integers, fractions, and irrational exponents. Multiply by the exponent, then reduce the exponent by one.

How do you differentiate polynomials?

Differentiate term by term using the power rule. Each term drops its degree by one; the constant term disappears. A degree-n polynomial has a degree-(n−1) derivative. After n differentiations, you get a constant; one more gives zero.

What are the derivatives of sine and cosine?

d/dx[sin x] = cos x and d/dx[cos x] = −sin x. The negative sign in the cosine derivative is essential. Repeated differentiation cycles with period four: sin → cos → −sin → −cos → sin.

What are the derivatives of tan, cot, sec, and csc?

d/dx[tan x] = sec²x, d/dx[cot x] = −csc²x, d/dx[sec x] = sec x tan x, d/dx[csc x] = −csc x cot x. Cofunctions (cos, cot, csc) carry negative signs in their derivatives; sin, tan, sec do not.

What is the derivative of e^x?

d/dx[eˣ] = eˣ. The natural exponential is unique in being its own derivative. For general base a: d/dx[aˣ] = aˣ ln a. When a = e, ln a = 1 and the factor disappears.

What are all the common derivative formulas?

The core formulas are: constants → 0, xⁿ → nxⁿ⁻¹, sin x → cos x, cos x → −sin x, tan x → sec²x, eˣ → eˣ, aˣ → aˣ ln a, ln x → 1/x. Combined with differentiation rules, these handle all standard functions.

Common Derivatives

Q: What is the derivative of ln x?

d/dx[ln x] = 1/x for x > 0. For general base a: d/dx[logₐ x] = 1/(x ln a). The natural logarithm gives the simplest derivative. The function ln|x| extends to all x ≠ 0 with derivative 1/x.

Constant Functions

Power Functions

Polynomials

Trigonometric Functions — Sine and Cosine

Trigonometric Functions — Tangent, Cotangent, Secant, Cosecant

Exponential Functions

Logarithmic Functions

Summary of Common Derivatives

The Core Derivative Library

A small collection of derivative formulas covers the vast majority of differentiation work. Polynomials, trigonometric functions, exponentials, and logarithms appear constantly, and their derivatives should be available from memory. Every more complex derivative—a composition, a product, an implicitly defined relation—ultimately reduces to these building blocks combined through differentiation rules.

Each formula below is provable from the limit definition. For some, the proof is a direct algebraic manipulation of the difference quotient. For others, it requires special limits such as

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

. The proofs confirm the formulas; the formulas replace the proofs in daily computation.

Constant Functions

For any constant

c

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

The graph of a constant function is a horizontal line. Every secant line through two points on it has slope zero, so the tangent line at every point also has slope zero. From the limit definition, the difference quotient

\frac{c - c}{h} = 0

for all

h \neq 0

, and the limit is

0

.

Under differentiation, constant terms vanish. In any sum

f(x) + c

, the constant contributes nothing to the derivative:

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

Power Functions

For any real exponent

n

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

The rule applies uniformly across exponent types. Positive integers:

\frac{d}{dx}[x^5] = 5x^4

. Negative integers:

\frac{d}{dx}[x^{-3}] = -3x^{-4}

. Fractions:

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

. Irrational exponents:

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

.

For positive integer

n

, the proof expands

(x+h)^n

using the binomial theorem. The leading term after cancellation is

nx^{n-1}h

, and dividing by

h

and taking the limit gives

nx^{n-1}

. All higher-order terms contain

h

as a factor and vanish in the limit.

Extension to negative and fractional exponents uses the quotient rule and the chain rule respectively, or alternatively logarithmic differentiation: writing

x^n = e^{n \ln x}

and applying the chain rule gives

\frac{d}{dx}[x^n] = e^{n \ln x} \cdot \frac{n}{x} = x^n \cdot \frac{n}{x} = nx^{n-1}

Polynomials

A polynomial

p(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0

is differentiated term by term using the power rule, constant multiple rule, and sum rule:

p'(x) = na_n x^{n-1} + (n-1)a_{n-1}x^{n-2} + \cdots + a_1

Each term drops its degree by one. The constant term

a_0

disappears. A polynomial of degree

n

has a derivative of degree

n - 1

.

Polynomials are differentiable at every real number. Their derivatives are again polynomials, so the process can be repeated indefinitely. After

n

differentiations, a degree-

n

polynomial becomes a constant

n! \cdot a_n

. One more differentiation yields zero, and all subsequent derivatives remain zero. This termination property distinguishes polynomials from transcendental functions, whose derivatives cycle or persist indefinitely.

Trigonometric Functions — Sine and Cosine

The two fundamental trigonometric derivatives are:

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

The sine derivative is proved from the limit definition. Expanding

\sin(x + h)

using the angle addition formula gives

\sin x \cos h + \cos x \sin h

. The difference quotient becomes

\frac{\sin x(\cos h - 1) + \cos x \sin h}{h} = \sin x \cdot \frac{\cos h - 1}{h} + \cos x \cdot \frac{\sin h}{h}

The special limits

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

and

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

yield the result:

\cos x

.

The cosine derivative follows similarly, or by differentiating

\cos x = \sin(\pi/2 - x)

using the chain rule. The negative sign in

(\cos x)' = -\sin x

is essential and a frequent source of error.

Repeated differentiation cycles with period four:

\sin x \to \cos x \to -\sin x \to -\cos x \to \sin x

. This periodicity extends to higher-order derivatives:

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

Trigonometric Functions — Tangent, Cotangent, Secant, Cosecant

The remaining four trigonometric derivatives follow from sine and cosine via the quotient rule or rewriting in terms of sine and cosine.

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

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

For

\tan x = \frac{\sin x}{\cos x}

, the quotient rule gives

\frac{\cos x \cdot \cos x - \sin x \cdot (-\sin x)}{\cos^2 x} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x

.

A pattern runs through the six derivatives: the cofunctions—

\cos

\cot

\csc

—each carry a negative sign in their derivatives, while

\sin

\tan

\sec

do not. This sign pattern is worth noting as a memory aid rather than a coincidence.

Each derivative is valid on the domain of the original function. The derivatives of

\tan x

and

\sec x

are undefined at

x = \pi/2 + k\pi

(where

\cos x = 0

). The derivatives of

\cot x

and

\csc x

are undefined at

x = k\pi

(where

\sin x = 0

Exponential Functions

The natural exponential function has the unique property of being its own derivative:

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

No other function satisfies

f'(x) = f(x)

except constant multiples

Ce^x

. This self-replicating property is what makes

e

the natural base for exponential functions.

The proof from the limit definition uses the special limit

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

. The difference quotient for

e^x

factors as

e^x \cdot \frac{e^h - 1}{h}

, and the limit gives

e^x \cdot 1 = e^x

.

For a general base

a > 0

a \neq 1

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

This follows from rewriting

a^x = e^{x \ln a}

and applying the chain rule. The factor

\ln a

is constant—it scales the derivative. When

a = e

\ln a = 1

and the factor disappears, confirming the special role of base

e

Logarithmic Functions

The natural logarithm has derivative:

\frac{d}{dx}[\ln x] = \frac{1}{x} \qquad x > 0

This can be derived by implicit differentiation. If

y = \ln x

, then

e^y = x

. Differentiating both sides:

e^y \frac{dy}{dx} = 1

, so

\frac{dy}{dx} = \frac{1}{e^y} = \frac{1}{x}

.

Alternatively, the limit definition gives

\lim_{h \to 0} \frac{\ln(x+h) - \ln x}{h} = \lim_{h \to 0} \frac{1}{h}\ln\left(\frac{x+h}{x}\right) = \lim_{h \to 0} \frac{1}{h}\ln\left(1 + \frac{h}{x}\right)

, which evaluates to

\frac{1}{x}

using the limit definition of

e

.

For a general base

a > 0

a \neq 1

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

This follows from the change of base formula

\log_a x = \frac{\ln x}{\ln a}

and the constant multiple rule. The natural logarithm gives the simplest derivative—another reason

e

is the preferred base.

The function

\ln|x|

extends the domain to all

x \neq 0

, and its derivative is

\frac{1}{x}

for both positive and negative

x

. This extended form appears frequently in integration.

Summary of Common Derivatives

The complete set of common derivative formulas:

\frac{d}{dx}[c] = 0 \qquad \frac{d}{dx}[x^n] = nx^{n-1}

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

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

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

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

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

These twelve formulas, combined with the differentiation rules, handle every explicit function built from powers, trigonometric functions, exponentials, and logarithms. Functions involving inverse trigonometric, hyperbolic, or piecewise definitions require the additional formulas collected in derivatives of special functions.