Summary of Differentiation Formulas

The formulas marked with a circle are, in my opinion, not worth memorizing. They are rarely used and fairly easy to derive if you need them.

Constants: $\displaystyle \frac{d}{dx}\left({c}\right) = 0$

Linear Functions: $\displaystyle \frac{d}{dx}\left({cx}\right) = c$

Monomials: $\displaystyle \frac{d}{dx}\left({x^n}\right) = nx^{n-1}$

Trig Functions:

  • $\displaystyle \frac{d}{dx}\left({\sin x}\right) = \cos x$
  • $\displaystyle \frac{d}{dx}\left({\cos x}\right) = -\sin x$
  • $\displaystyle \frac{d}{dx}\left({\tan x}\right) = \sec^2 x$
  • $\displaystyle \frac{d}{dx}\left({\csc x}\right) = -\cot x \csc x$
  • $\displaystyle \frac{d}{dx}\left({\sec x}\right) = \tan x \sec x$
  • $\displaystyle \frac{d}{dx}\left({\cot x}\right) = -\csc^2 x$

Inverse Trig Functions:

  • $\displaystyle \frac{d}{dx}\left({\sin^{-1} x}\right) = \frac{1}{\sqrt{1-x^2}}$
  • $\displaystyle \frac{d}{dx}\left({\cos^{-1} x}\right) = -\frac{1}{\sqrt{1-x^2}}$
  • $\displaystyle \frac{d}{dx}\left({\tan^{-1} x}\right) = \frac{1}{1+x^2}$
  • $\displaystyle \frac{d}{dx}\left({\csc^{-1} x}\right) = -\left\vert\frac{1}{x\sqrt{x^2-1}}\right\vert$
  • $\displaystyle \frac{d}{dx}\left({\sec^{-1} x}\right) = \left\vert\frac{1}{x\sqrt{x^2-1}}\right\vert$
  • $\displaystyle \frac{d}{dx}\left({\cot^{-1} x}\right) = -\frac{1}{1+x^2}$

Exponential Functions:

  • $\displaystyle \frac{d}{dx}\left({e^x}\right) = e^x$
  • $\displaystyle \frac{d}{dx}\left({b^x}\right) = b^x\ln b$

Logarithmic Functions:

  • $\displaystyle \frac{d}{dx}\left({\ln x}\right) = \frac{1}{x}$
  • $\displaystyle \frac{d}{dx}\left({\log_b x}\right) = \frac{1}{x\ln b}$

The Constant Multiple Rule: $\displaystyle \frac{d}{dx}\left({cf(x)}\right) = c\frac{df(x)}{dx}$

The Sum Rule: $\displaystyle \frac{d}{dx}\left({f(x)+g(x)}\right) = \frac{df(x)}{dx}+\frac{dg(x)}{dx}$

The Difference Rule: $\displaystyle \frac{d}{dx}\left({f(x)-g(x)}\right) = \frac{df(x)}{dx}-\frac{dg(x)}{dx}$

The Product Rule: $\displaystyle \frac{d}{dx}\left({f(x)g(x)}\right) = g(x)\frac{df(x)}{dx}+f(x)\frac{dg(x)}{dx}$

The Quotient Rule: $\displaystyle \frac{d}{dx}\left({\!\frac{f(x)}{g(x)}\!}\right) = \frac{g(x)\frac{df(x)}{dx}-f(x)\frac{dg(x)}{dx}}{g(x)^2}$

The Chain Rule: $\displaystyle \frac{d}{dx}\left({f\circ g)(x)}\right) = \frac{df(g(x))}{dx}\,\cdot\,\frac{dg(x)}{dx}$

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