Derivative/Formulas

List of formulas

 $\frac d{dx}(cf(x)) = c\left(\frac d{dx} f(x)\right)$ where $c$ is a constant $(f(x) + g(x))' = f'(x) + g'(x)$ $(f(x)-g(x))'=f'(x)-g'(x)$ $\left(u(x)\times v(x)\right)'=u(x)v'(x)+u'(x)v(x)$ $\left(\frac{u(x)}{v(x)}\right)' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ $(f(g(x)))' = f'(g(x))g'(x)$ $\frac d{dx} x^n = n x^{n-1}$ $\frac d{dx} (f(x))^n =n f(x)^{n-1} f'(x)$ $\frac d{dx} \sin x = \cos x$ $\frac d{dx} \cos x = -\sin x$ $\frac d{dx} \tan x = \sec^2 x$ $\frac d{dx} \sec x = \sec x \tan x$ $\frac d{dx} \csc x = -\csc x\cot x$ $\frac d{dx} \cot x = -\csc^2 x$ $\frac d{dx} e^x = e^x$ $\frac d{dx} a^x = (\ln a) a^x$ $\frac d{dx} \ln x = \frac 1x$ $\frac d{dx} \log_b x =\frac{\log_b e}{x}$ $\frac d{dx} \arcsin x = \frac 1{\sqrt{1-x^2}}$ $\frac d{dx} \arccos x = -\frac 1{\sqrt{1-x^2}}$ $\frac d{dx} \arctan x = \frac 1{1+x^2}$ $\frac d{dx} \mathrm{arcsec \ } x = \frac 1{\lvert x \rvert \sqrt{x^2-1}}$ $\frac d{dx} \mathrm{arccsc \ } x = - \frac 1{x\sqrt{x^2 - 1}}$ $\frac d{dx} \mathrm{arccot \ } x = - \frac 1{1+x^2}$

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 $\ddot{x}$