# Calculus/Tables of Derivatives

## General Rules

$\frac{d}{dx}(f + g)= \frac{df}{dx} + \frac{dg}{dx}$

$\frac{d}{dx}(cf)= c\frac{df}{dx}$

$\frac{d}{dx}(fg)= f\frac{dg}{dx} + g\frac{df}{dx}$

$\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{g\frac{df}{dx} - f\frac{dg}{dx}}{g^2}$

$[f(g(x))]' = f'(g(x)) g'(x)$

## Powers and Polynomials

• $\frac{d}{dx} (c) = 0$
• $\frac{d}{dx}x=1$
• $\frac{d}{dx}x^n=nx^{n-1}$
• $\frac{d}{dx}\sqrt{x}=\frac{1}{2\sqrt x}$
• $\frac{d}{dx}\frac{1}{x}=-\frac{1}{x^2}$
• $\frac{d}{dx}(c_n x^n + c_{n-1} x^{n-1} + c_{n-2}x^{n-2} + \cdots +c_2x^2 + c_1 x + c_0) = n c_n x^{n-1} + (n-1) c_{n-1} x^{n-2} + (n-2) c_{n-2}x^{n-3} + \cdots + 2c_2x+ c_1$

## Trigonometric Functions

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

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

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

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

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

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

## Exponential and Logarithmic Functions

• $\frac{d}{dx} e^x =e^x$
• $\frac{d}{dx} a^x =a^x \ln (a)\qquad\mbox{if }a>0$
• $\frac{d}{dx} \ln (x)= \frac{1}{x}$
• $\frac{d}{dx} \log_a (x)= \frac{1}{x\ln (a)}\qquad\mbox{if }a>0, a\neq 1$
• $(f^g)' = \left(e^{g\ln f}\right)' = f^g\left(f'{g \over f} + g'\ln f\right),\qquad f > 0$
• $(c^f)' = \left(e^{f\ln c}\right)' = f' c^f \ln c$

## Inverse Trigonometric Functions

• $\frac{d}{dx} \mbox{arcsin x} = \frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx} \mbox{arccos x} = -\frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx} \mbox{arctan x} = \frac{1}{1+x^2}$
• ${d \over dx} \arcsec x = { 1 \over |x|\sqrt{x^2 - 1}}$
• ${d \over dx} \arccot x = {-1 \over 1 + x^2}$
• ${d \over dx} \arccsc x = {-1 \over |x|\sqrt{x^2 - 1}}$

## Hyperbolic and Inverse Hyperbolic Functions

${d \over dx} \sinh x = \cosh x$
${d \over dx} \cosh x = \sinh x$
${d \over dx} \tanh x = \mbox{sech}^2\,x$
${d \over dx} \,\mbox{sech}\,x = -\tanh x\,\mbox{sech}\,x$
${d \over dx} \,\mbox{coth}\,x = -\,\mbox{csch}^2\,x$
${d \over dx} \,\mbox{csch}\,x = -\,\mbox{coth}\,x\,\mbox{csch}\,x$
${d \over dx} \sinh^{-1} x = { 1 \over \sqrt{x^2 + 1}}$
${d \over dx} \cosh^{-1} x = {-1 \over \sqrt{x^2 - 1}}$
${d \over dx} \tanh^{-1} x = { 1 \over 1 - x^2}$
${d \over dx} \mbox{sech}^{-1}\,x = { 1 \over x\sqrt{1 - x^2}}$
${d \over dx} \mbox{coth}^{-1}\,x = {-1 \over 1 - x^2}$
${d \over dx} \mbox{csch}^{-1}\,x = {-1 \over |x|\sqrt{1 + x^2}}$