Calculus reference guide

Common derivatives to memorize

The core formulas calculus students are usually expected to recall quickly, organized by function family and rule.

Angles are in radiansPrimes indicate derivativesDomain notes are included where useful

Start here

What should I memorize first?

Begin with the power rule, the derivatives of sine and cosine, the natural exponential and logarithm, and the product, quotient, and chain rules. Add the remaining trig formulas next. Inverse trig derivatives are important, but they are usually a later priority.

Core algebraic rules

Memorize these first. They appear in almost every differentiation problem.

Constant rule

The derivative of a constant is zero.

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

Power rule

Valid wherever the original power function is differentiable.

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

Constant multiple rule

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

Sum and difference rule

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

Exponential and logarithmic functions

The natural exponential and natural logarithm form the essential pair.

Natural exponential

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

Exponential with base a

Assume a > 0 and a ≠ 1.

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

Natural logarithm

For x > 0.

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

Natural log of absolute x

For x ≠ 0.

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

Logarithm with base a

Assume x > 0, a > 0, and a ≠ 1.

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

Trigonometric functions

These formulas assume angles are measured in radians.

Sine

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

Cosine

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

Tangent

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

Cotangent

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

Secant

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

Cosecant

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

Inverse trigonometric functions

The first three are the most frequently used; arcsecant and arccosecant require absolute values.

Inverse sine

For |x| < 1.

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

Inverse cosine

For |x| < 1.

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

Inverse tangent

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

Inverse cotangent

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

Inverse secant

For |x| > 1.

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

Inverse cosecant

For |x| > 1.

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

Rules for combining functions

These tell you what to do when functions are multiplied, divided, or nested.

Product rule

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

Quotient rule

Assume g(x) ≠ 0.

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

Chain rule

Differentiate the outside, keep the inside, then multiply by the inside derivative.

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

Common chain-rule forms

Let u be a differentiable function of x. These are the memorized formulas with the chain rule already visible.

Power of a function

\frac{d}{dx}[u^n]=nu^{n-1}u'

Natural exponential of a function

\frac{d}{dx}\left[e^u\right]=e^u u'

Exponential of a function

\frac{d}{dx}\left[a^u\right]=a^u\ln(a)\,u'

Natural log of a function

Where u ≠ 0.

\frac{d}{dx}[\ln|u|]=\frac{u'}{u}

Sine of a function

\frac{d}{dx}[\sin u]=\cos(u)\,u'

Cosine of a function

\frac{d}{dx}[\cos u]=-\sin(u)\,u'

Tangent of a function

\frac{d}{dx}[\tan u]=\sec^2(u)\,u'

One important reminder

Memorizing the table is only the first step.

Most real problems combine these formulas. Before differentiating, identify whether the expression is a sum, product, quotient, or composition. Then choose the outer rule first and work inward.