# Derivative

The derivative of a function is defined as the instantaneous rate of change of the function with respect to one of the variables.

## Notation

The following are commonly recognized notations for expressing the derivative of a function.

 Euler's notation First derivative $D_xf(x)$ or $Du$ Second derivative $D_x^2f(x)$ or $D^2u$ Third derivative $D_x^3f(x)$ or $D^3u$ $n$th derivative $D_x^nf(x)$ or $D^nu$ Lagrange's notation First derivative $f'(x)$ Second derivative $f''(x)$ Third derivative $f'''(x)$ $n$th derivative $f^{(n)}(x)$ Leibniz's notation First derivative $\frac{dy}{dx}$ Second derivative $\frac{d^2y}{dx^2}$ $n$th derivative $\frac{d^ny}{dx^n}$ Newton's notation First derivative $\dot{x}$ Second derivative $\ddot{x}$