Common derivatives

This page presents the most used derivatives gathered in the following categories:

Derivatives of polynomial functions
Derivatives of trigonometric functions
derivatives of inverse trigonometric functions
Derivatives of exponential functions
Derivatives of logarithm functions
Derivatives of hyperbolic trigonometric functions

Polynomials

Polynomials are one of the simplest functions to differentiate. Derivative of polynomialsby is mostly based on the power rule:

$$ \frac{d}{dx}(c) = 0 \notag $$

$$ \frac{d}{dx}(x)= 1 \notag $$

$$ \frac{d}{dx}(ax) = a \notag $$

$$ \frac{d}{dx}(x^n) = nx^{n-1} \notag $$

$$ \frac{d}{dx}(cx^n) = ncx^{n-1} \notag $$

$$ \frac{d}{dx}(ax^2 + bx + c) = 2ax + b \notag $$

Trigonometric functions

The differentiation of trigonometric functions can be done using the derivatives of sin(x) and cos(x) and applying the quotient rule. The differentiation formulas of the six trigonometric functions are listed below:

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

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

$$ \frac{d}{dx} \tan(x) = \sec^2(x) \notag $$

$$ \frac{d}{dx} \sec(x) = \sec (x) \times tan(x) \notag $$

$$ \frac{d}{dx} \csc(x) = -csc(x) \times cot(x) \notag $$

$$ \frac{d}{dx} \cot(x) = -csc^2(x) \notag $$

Inverse trigonometric functions

$$ \frac{d}{dx} \sin^{-1}(x) = \frac {1} { \sqrt { 1-x^2 } } \notag $$

$$ \frac{d}{dx} \cos^{-1}(x) = - \frac {1} { \sqrt { 1-x^2 } } \notag $$

$$ \frac{d}{dx} \tan^{-1}(x) = \frac {1} { 1 + x^2 } \notag $$

$$ \frac{d}{dx} \sec^{-1}(x) = \frac {1} { \left| x \right| \sqrt { x^2 - 1 } } \notag $$

$$ \frac{d}{dx} \csc^{-1}(x) = - \frac {1} { \left| x \right| \sqrt { x^2 -1 } } \notag $$

$$ \frac{d}{dx} \cot^{-1}(x) = - \frac {1} { 1+x^2 } \notag $$

Exponential functions

$$ \frac{d}{dx} (a^x) = \ln(a) \times a^x \notag $$

$$ \frac{d}{dx} (\mathrm{e}^x) = \mathrm{e}^x \notag $$

Logarithm functions

$$ \frac{d}{dx} \ln(x) = \frac{1}{x} \quad \forall \quad x>0 \notag $$

$$ \frac{d}{dx} \ln( \left| x \right| ) = \frac{1}{x} \quad \forall \quad x \neq 0 \notag $$

$$ \frac{d}{dx} \log_n(x) = \frac{1}{x \ln n} \quad \forall \quad x>0 \notag $$

Hyperbolic trigonometric functions

$$ \frac{d}{dx} \sinh(x) = \cosh(x) \notag $$

$$ \frac{d}{dx} \cosh(x) = \sinh(x) \notag $$

$$ \frac{d}{dx} \tanh(x) = \mathrm{sech}^2(x) \notag $$

$$ \frac{d}{dx} \mathrm{ sech} (x) = - \mathrm{ sech(x) } \times \tanh(x) \notag $$

$$ \frac{d}{dx} \mathrm{ csch }(x) = - \mathrm{ csch(x) } \times coth(x) \notag $$

$$ \frac{d}{dx} \coth(x) = - \mathrm{ csch^2(x) } \notag $$