Derivative/Formulas

List of formulas

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

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 $\ddot{x}$

See also

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