# Calculus/Quick Reference

This quick reference lists formulas often used in solving calculus problems.

## Differentiation

Wikipedia includes a useful summary of differentiation rules. These include rules for computing the derivatives of the sum, product, and other combinations of functions.

### Rules for basic functions

Here are the rules for the derivatives of the most common basic functions, where a is a real number.

• Derivatives of powers:
${\displaystyle {\frac {d}{dx}}x^{a}=ax^{a-1}.}$
• Exponential and logarithmic functions:
${\displaystyle {\frac {d}{dx}}e^{x}=e^{x}.}$
${\displaystyle {\frac {d}{dx}}a^{x}=a^{x}\ln(a),\qquad a>0}$
${\displaystyle {\frac {d}{dx}}\ln(x)={\frac {1}{x}},\qquad x>0.}$
${\displaystyle {\frac {d}{dx}}\log _{a}(x)={\frac {1}{x\ln(a)}},\qquad x,a>0}$
• Trigonometric functions:
${\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x).}$
${\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x).}$
${\displaystyle {\frac {d}{dx}}\tan(x)=\sec ^{2}(x)={\frac {1}{\cos ^{2}(x)}}=1+\tan ^{2}(x).}$
• Inverse trigonometric functions:
${\displaystyle {\frac {d}{dx}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}},\qquad -1
${\displaystyle {\frac {d}{dx}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}},\qquad -1
${\displaystyle {\frac {d}{dx}}\arctan(x)={\frac {1}{1+x^{2}}}}$