Question 1

What are common derivatives?

Accepted Answer

Common derivatives are the derivative identities that come up most often in calculus: derivatives of constants, polynomials, linear functions, exponentials, logarithms, the six trigonometric functions, and the six inverse trigonometric functions. Together they cover the building blocks needed for almost any differentiation problem when combined with the linearity, product, quotient, and chain rules.

Question 2

What is the derivative of sin x and cos x?

Accepted Answer

The derivative of sin(x) is cos(x). The derivative of cos(x) is negative sin(x). The sign change on cosine reflects the 90-degree phase shift between the two functions. Repeatedly differentiating cycles through sin, cos, negative sin, negative cos, and back to sin every four steps.

Question 3

What is the power rule?

Accepted Answer

The power rule says that the derivative of x to the n is n times x to the (n minus 1) for any real exponent n. To use it, multiply by the exponent and then subtract one from the exponent. The rule covers polynomials, roots like the square root of x, and reciprocals like 1 over x.

Question 4

What is the difference between the product rule and the chain rule?

Accepted Answer

The product rule differentiates a product of two functions: the derivative of f times g is f-prime times g plus f times g-prime. The chain rule differentiates a composition: the derivative of f of g(x) is f-prime evaluated at g(x), times g-prime of x. Products combine functions side by side; compositions nest one function inside another.

Question 5

How do I memorize derivative rules?

Accepted Answer

Group the identities by category: polynomial, linear, exponential, logarithmic, trigonometric, and inverse trigonometric. Notice the patterns: cosine differentiates to negative sine; tangent and cotangent share a sign-flip with secant squared and cosecant squared; arcsin and arccos have opposite signs because they sum to a constant. Drill with the drag puzzle on this page to lock the pairings in.

Name	f(x)		f′(x)	Family
Constant rule	$\frac{d}{dx}\!\left[c\right]$	=	$0$	Polynomial
Identity rule	$\frac{d}{dx}\!\left[x\right]$	=	$1$	Polynomial
Linear rule	$\frac{d}{dx}\!\left[ax + b\right]$	=	$a$	Linear
Power rule	$\frac{d}{dx}\!\left[x^n\right]$	=	$nx^{n-1}$	Polynomial
Reciprocal	$\frac{d}{dx}\!\left[\frac{1}{x}\right]$	=	$-\frac{1}{x^2}$	Polynomial
Square root	$\frac{d}{dx}\!\left[\sqrt{x}\right]$	=	$\frac{1}{2\sqrt{x}}$	Polynomial
Natural exponential	$\frac{d}{dx}\!\left[e^x\right]$	=	$e^x$	Exponential
General exponential	$\frac{d}{dx}\!\left[a^x\right]$	=	$a^x \ln(a)$	Exponential
Natural logarithm	$\frac{d}{dx}\!\left[\ln(x)\right]$	=	$\frac{1}{x}$	Logarithmic
General logarithm	$\frac{d}{dx}\!\left[\log_a(x)\right]$	=	$\frac{1}{x \ln(a)}$	Logarithmic
Sine	$\frac{d}{dx}\!\left[\sin(x)\right]$	=	$\cos(x)$	Trigonometry
Cosine	$\frac{d}{dx}\!\left[\cos(x)\right]$	=	$-\sin(x)$	Trigonometry
Tangent	$\frac{d}{dx}\!\left[\tan(x)\right]$	=	$\sec^2(x)$	Trigonometry
Cotangent	$\frac{d}{dx}\!\left[\cot(x)\right]$	=	$-\csc^2(x)$	Trigonometry
Secant	$\frac{d}{dx}\!\left[\sec(x)\right]$	=	$\sec(x)\tan(x)$	Trigonometry
Cosecant	$\frac{d}{dx}\!\left[\csc(x)\right]$	=	$-\csc(x)\cot(x)$	Trigonometry
Arcsine	$\frac{d}{dx}\!\left[\arcsin(x)\right]$	=	$\frac{1}{\sqrt{1-x^2}}$	Inverse trigonometry
Arccosine	$\frac{d}{dx}\!\left[\arccos(x)\right]$	=	$-\frac{1}{\sqrt{1-x^2}}$	Inverse trigonometry
Arctangent	$\frac{d}{dx}\!\left[\arctan(x)\right]$	=	$\frac{1}{1+x^2}$	Inverse trigonometry
Arccotangent	$\frac{d}{dx}\!\left[\operatorname{arccot}(x)\right]$	=	$-\frac{1}{1+x^2}$	Inverse trigonometry
Arcsecant	$\frac{d}{dx}\!\left[\operatorname{arcsec}(x)\right]$	=	$\frac{1}{\|x\|\sqrt{x^2-1}}$	Inverse trigonometry
Arccosecant	$\frac{d}{dx}\!\left[\operatorname{arccsc}(x)\right]$	=	$-\frac{1}{\|x\|\sqrt{x^2-1}}$	Inverse trigonometry

Common Derivatives

Derivative Tool

Categories

Polynomial

Linear

Exponential

Logarithmic

Trigonometry

Inverse trigonometry

Reciprocal forms

Differentiation rules

Linearity

Product rule

Quotient rule

Chain rule