Derivatives and integrals of basic functions

Functions Derivatives Indefinite integrals
Fundamental theorem

of calculus

   
  $ \displaystyle\frac{d}{dx} F(x) = f(x)$ $ \displaystyle\int f(x) dx = F(x) + C$
Power and logarithmic

functions

   
  $ \displaystyle\frac{d}{dx} x^n = n x^{n-1}$ $ \displaystyle\int x^n dx = \frac{x^{n+1}}{n+1} + C$ ($ n \neq -1$)
  $ \displaystyle\frac{d}{dx} \ln \vert x\vert = \frac{1}{x}$ $ \displaystyle\int \frac{1}{x} dx = \ln \vert x\vert + C$
Exponential functions    
  $ \displaystyle\frac{d}{dx} e^x = e^x$ $ \displaystyle\int e^x dx = e^x + C$
Trigonometric functions    
  $ \displaystyle\frac{d}{dx} \sin x = \cos x$ $ \displaystyle\int \cos x dx = \sin x + C$
  $ \displaystyle\frac{d}{dx} \cos x = -\sin x$ $ \displaystyle\int \sin x dx = -\cos x + C$
  $ \displaystyle\frac{d}{dx} \tan x = \frac{1}{\cos^2 x} = \sec^2 x$ $ \displaystyle\int \sec^2 x dx =
\int \frac{1}{\cos^2 x} dx = \tan x + C$
  $ \displaystyle\frac{d}{dx} \cot x = -\frac{1}{\sin^2 x} = -\csc^2 x$ $ \displaystyle\int \csc^2 x dx =
\int \frac{1}{\sin^2 x} dx = -\cot x + C$
  $ \displaystyle\frac{d}{dx} \sec x = \sec x \tan x$ $ \displaystyle\int \sec x \tan x dx = \sec x + C$
Hyperbolic functions    
  $ \displaystyle\frac{d}{dx} \sinh x = \cosh x$ $ \displaystyle\int \cosh x dx = \sinh x + C$
  $ \displaystyle\frac{d}{dx} \cosh x = \sinh x$ $ \displaystyle\int \sinh x dx = \cosh x + C$
  $ \displaystyle\frac{d}{dx} \tanh x = \frac{1}{\cosh^2 x}$ $ \displaystyle\int \frac{1}{\cosh^2 x} dx = \tanh x + C$
  $ \displaystyle\frac{d}{dx} \coth x = -\frac{1}{\sinh^2 x}$ $ \displaystyle\int \frac{1}{\sinh^2 x} dx = -\coth x + C$
Inverse trigonometric

functions

   
  $ \displaystyle\frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1 - x^2}}$ $ \displaystyle\int \frac{1}{\sqrt{1 - x^2}} dx = \sin^{-1} x + C$
  $ \displaystyle\frac{d}{dx} \cos^{-1} x = -\frac{1}{\sqrt{1 - x^2}}$  
  $ \displaystyle\frac{d}{dx} \tan^{-1} x = \frac{1}{1 + x^2}$ $ \displaystyle\int \frac{1}{1 + x^2} dx = \tan^{-1} x + C$
Inverse hyperbolic

functions

   
  $ \displaystyle\frac{d}{dx} \sinh^{-1} x = \frac{1}{\sqrt{1 + x^2}}$ $ \displaystyle\int \frac{1}{\sqrt{1 + x^2}} dx = \sinh^{-1} x + C$
  $ \displaystyle\frac{d}{dx} \cosh^{-1} x = \frac{1}{\sqrt{x^2 - 1}}$ $ \displaystyle\int \frac{1}{\sqrt{x^2 - 1}} dx = \cosh^{-1} x + C$
  $ \displaystyle\frac{d}{dx} \tanh^{-1} x = \frac{1}{1 - x^2}$ $ \displaystyle\int \frac{1}{1 - x^2} dx = \tanh^{-1} x + C$



Department of Mathematics
Last modified: 2005-10-05