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:

  $\frac{d}{dx}{x}^a=a{x}^{a-1}.$

• Exponential and logarithmic functions:

  $\frac{d}{dx}{e}^x=e^x.$

  $\frac{d}{dx}{a}^x=a^x\ln (a),a>0$

  $\frac{d}{dx}\ln (x)=\frac{1}{x},x>0.$

  $\frac{d}{dx}{\log }_a(x)=\frac{1}{x\ln (a)},x,a>0$

• Trigonometric functions:

  $\frac{d}{dx}\sin (x)=\cos (x).$

  $\frac{d}{dx}\cos (x)=-\sin (x).$

  $\frac{d}{dx}\tan (x)={\sec }^2(x)=\frac{1}{{\cos }^2(x)}=1+{\tan }^2(x).$

• Inverse trigonometric functions:

  $\frac{d}{dx}\arcsin (x)=\frac{1}{\sqrt{1-x^2}},-1<x<1.$

  $\frac{d}{dx}\arccos (x)=-\frac{1}{\sqrt{1-x^2}},-1<x<1.$

  $\frac{d}{dx}\arctan (x)=\frac{1}{1+x^2}$