Derivative

Derivative show the amount by which a function is changing at one given point. It's particularly used to calculate the slope of a function.

The partial derivative of a function is the of one named variable, where all the other ones are held constant.

Notation

	Lagrange	Leibniz
Function	$f(x)$	$f$
1st derivative	$f^{\prime }(x)$	$\frac{df}{dx}$
2nd derivative	$f^{″}(x)$	$\frac{d^{2}f}{d{x}^2}$
$n^{\text{th}}$ derivative	$f^{(n)}(x)$	$\frac{d^{n}f}{d{x}^n}$

Table

$y=f(x)$	$\frac{dy}{dx}=f^{\prime }(x)$
$k$, any constant	$0$
$x$	$1$
$x^2$	$2x$
$x^3$	$3x^2$
$x^n$, any constant $n$	$n{x}^{n-1}$
$e^x$	$e^x$
$e^{kx}$	$k{e}^{kx}$
$\ln x={\log }_{e}x$	$\frac{1}{x}$
$\sin x$	$\cos x$
$\sin kx$	$k\cos kx$
$\cos x$	$-\sin x$
$\cos kx$	$-k\sin kx$
$\tan x=\frac{\sin x}{\cos x}$	${\sec }^{2}x$
$\tan kx$	$k{\sec }^{2}kx$
$\csc x=\frac{1}{\sin x}$	$-\csc x\cot x$
$\sec x=\frac{1}{\cos x}$	$\sec x\tan x$
$\cot x=\frac{\cos x}{\sin x}$	$-{\csc }^{2}x$
${\sin }^{-1}x$	$\frac{1}{\sqrt{1-x^2}}$
${\cos }^{-1}x$	$-\frac{1}{\sqrt{1-x^2}}$
${\tan }^{-1}x$	$\frac{1}{1+x^2}$
$\cosh x$	$\sinh x$
$\sinh x$	$\cosh x$
$\tanh x$	${\text{sech}}^{²}x$
$\text{sech}x=\frac{1}{\cosh x}$	$-\text{sech}x\tanh x$
$\text{cosech}x=\frac{1}{\sinh x}$	$-\text{cosech}x\coth x$
$\coth x=\frac{\cosh x}{\sinh x}$	$-{\text{cosech}}^{2}x$
${\cosh }^{-1}x$	$\frac{1}{\sqrt{x^2-1}}$
${\sinh }^{-1}x$	$\frac{1}{\sqrt{x^2+1}}$
${\tanh }^{-1}x$	$\frac{1}{1-x^2}$

Matrix

When a matrix is composed of derivatives, it called:

• Jacobian-Matrix, for first-order derivatives
• Hessian matrix, for second-order derivatives