Calculus/Tables of Derivatives

General Rules

$\frac{\mathrm{d}}{\mathrm{d}x}(f+g)=\frac{\mathrm{d}f}{\mathrm{d}x}+\frac{\mathrm{d}g}{\mathrm{d}x}$

$\frac{\mathrm{d}}{\mathrm{d}x}(c\cdot f)=c\cdot \frac{\mathrm{d}f}{\mathrm{d}x}$

$\frac{\mathrm{d}}{\mathrm{d}x}(f\cdot g)=f\cdot \frac{\mathrm{d}g}{\mathrm{d}x}+g\cdot \frac{\mathrm{d}f}{\mathrm{d}x}$

$\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{f}{g}\right)=\frac{-f\cdot \frac{\mathrm{d}g}{dx}+g\cdot \frac{\mathrm{d}f}{\mathrm{d}x}}{g^2}$

$\frac{\mathrm{d}}{\mathrm{d}x}[f(g(x))]=\frac{\mathrm{d}f}{\mathrm{d}g}\cdot \frac{\mathrm{d}g}{\mathrm{d}x}=f^{\prime }(g(x))\cdot g^{\prime }(x)$

$\frac{{\mathrm{d}}^n}{\mathrm{d}{x}^n}f(x)g(x)=\sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)f^{(n-i)}(x)g^{(i)}(x)$

$\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{1}{f}\right)=-\frac{f^{\prime }}{f^2}$

Powers and Polynomials

• $\frac{\mathrm{d}}{\mathrm{d}x}(c)=0$
• $\frac{\mathrm{d}}{\mathrm{d}x}x=1$
• $\frac{\mathrm{d}}{\mathrm{d}x}{x}^n=n{x}^{n-1}$
• $\frac{\mathrm{d}}{\mathrm{d}x}\sqrt{x}=\frac{1}{2\sqrt{x}}$
• $\frac{\mathrm{d}}{\mathrm{d}x}\frac{1}{x}=-\frac{1}{x^2}$
• $\frac{\mathrm{d}}{\mathrm{d}x}(c_n{x}^n+c_{n-1}{x}^{n-1}+c_{n-2}{x}^{n-2}+\cdots +c_2{x}^2+c_{1}x+c_0)=n{c}_n{x}^{n-1}+(n-1)c_{n-1}{x}^{n-2}+(n-2)c_{n-2}{x}^{n-3}+\cdots +2c_{2}x+c_1$

Trigonometric Functions

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

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

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

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

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

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

Exponential and Logarithmic Functions

• $\frac{\mathrm{d}}{\mathrm{d}x}{e}^x=e^x$
• $\frac{\mathrm{d}}{\mathrm{d}x}{a}^x=a^x\ln (a)\text{if }a>0$
• $\frac{\mathrm{d}}{\mathrm{d}x}\ln (x)=\frac{1}{x}$
• $\frac{\mathrm{d}}{\mathrm{d}x}{\log }_a(x)=\frac{1}{x\ln (a)}\text{if }a>0,a\ne 1$
• $\frac{\mathrm{d}}{\mathrm{d}x}(f^g)=\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{g\ln (f)}\right)=f^g(f^{\prime }\frac{g}{f}+g^{\prime }\ln (f)),f>0$
• $\frac{\mathrm{d}}{\mathrm{d}x}(c^f)=\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{f\ln (c)}\right)=c^f\ln (c)\cdot f^{\prime }$

Inverse Trigonometric Functions

$\frac{\mathrm{d}}{\mathrm{d}x}\arcsin (x)=\frac{1}{\sqrt{1-x^2}}$

$\frac{\mathrm{d}}{\mathrm{d}x}\arccos (x)=-\frac{1}{\sqrt{1-x^2}}$

$\frac{\mathrm{d}}{\mathrm{d}x}\arctan (x)=\frac{1}{x^2+1}$

$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{arccot}(x)=-\frac{1}{x^2+1}$

$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{arcsec}(x)=\frac{1}{|x|\sqrt{x^2-1}}$

$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{arccsc}(x)=-\frac{1}{|x|\sqrt{x^2-1}}$

Hyperbolic and Inverse Hyperbolic Functions

$\frac{\mathrm{d}}{\mathrm{d}x}\sinh (x)=\cosh (x)$

$\frac{\mathrm{d}}{\mathrm{d}x}\cosh (x)=\sinh (x)$

$\frac{\mathrm{d}}{\mathrm{d}x}\tanh (x)={\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2(x)$

$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(x)=-\tanh (x)\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(x)$

$\frac{\mathrm{d}}{\mathrm{d}x}\coth (x)=-{\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}}^2(x)$

$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}(x)=-\coth (x)\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}(x)$

$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(x)=\frac{1}{\sqrt{x^2+1}}$

$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(x)=\frac{1}{\sqrt{x^2-1}},x>1$

$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}(x)=\frac{1}{1-x^2},|x|<1$

$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(x)=-\frac{1}{x\sqrt{1-x^2}},0<x<1$

$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}(x)=\frac{1}{1-x^2},|x|>1$

$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}(x)=-\frac{1}{|x|\sqrt{1+x^2}},x\ne 0$

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