Calculus/Tables of Derivatives

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

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

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

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

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

$$\frac{\mathrm{d}^n}{\mathrm{d}x^n} f(x)g(x) = \sum_{i=0}^n \left(\begin{matrix}n\\i\end{matrix}\right)f^{(n-i)}(x)g^{(i)}(x)$$

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

Powers and Polynomials

 * $$\frac{\mathrm{d}}{\mathrm{d}x}(c)=0$$
 * $$\frac{\mathrm{d}}{\mathrm{d}x}x=1$$
 * $$\frac{\mathrm{d}}{\mathrm{d}x}x^n=nx^{n-1}$$
 * $$\frac{\mathrm{d}}{\mathrm{d}x}\sqrt{x}=\frac{1}{2\sqrt x}$$
 * $$\frac{\mathrm{d}}{\mathrm{d}x}\frac{1}{x}=-\frac{1}{x^2}$$
 * $${\frac{\mathrm{d}}{\mathrm{d}x}(c_nx^n+c_{n-1}x^{n-1}+c_{n-2}x^{n-2}+\cdots+c_2x^2+c_1x+c_0)=nc_nx^{n-1}+(n-1)c_{n-1}x^{n-2}+(n-2)c_{n-2}x^{n-3}+\cdots+2c_2x+c_1}$$

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

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

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

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

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

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

Exponential and Logarithmic Functions

 * $$\frac{\mathrm{d}}{\mathrm{d}x}e^x=e^x$$
 * $$\frac{\mathrm{d}}{\mathrm{d}x}a^x=a^x\ln(a)\qquad\text{if }a>0$$
 * $$\frac{\mathrm{d}}{\mathrm{d}x}\ln(x)=\frac{1}{x}$$
 * $$\frac{\mathrm{d}}{\mathrm{d}x}\log_a(x)=\frac{1}{x\ln(a)}\qquad\text{if }a>0\ ,\ a\ne1$$
 * $$\frac{\mathrm{d}}{\mathrm{d}x}(f^g)=\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{g\ln(f)}\right)=f^g\left(f'\frac{g}{f}+g'\ln(f)\right)\ ,\qquad f>0$$
 * $$\frac{\mathrm{d}}{\mathrm{d}x}(c^f)=\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{f\ln(c)}\right)=c^f\ln(c)\cdot f'$$

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

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

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

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

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

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

Hyperbolic and Inverse Hyperbolic Functions
$$\frac{\mathrm{d}}{\mathrm{d}x}\sinh(x)=\cosh(x)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}\tanh(x)={\rm sech}^2(x)$$

$$\frac{\mathrm{d}}{\mathrm{d}x}{\rm sech}(x)=-\tanh(x){\rm sech}(x)$$

$$\frac{\mathrm{d}}{\mathrm{d}x}\coth(x)=-{\rm csch}^2(x)$$

$$\frac{\mathrm{d}}{\mathrm{d}x}{\rm csch}(x)=-\coth(x){\rm csch}(x)$$

$$\frac{\mathrm{d}}{\mathrm{d}x}{\rm arsinh}(x)=\frac{1}{\sqrt{x^2+1}}$$

$$\frac{\mathrm{d}}{\mathrm{d}x}{\rm arcosh}(x)=\frac{1}{\sqrt{x^2-1}}\ ,\ x>1$$

$$\frac{\mathrm{d}}{\mathrm{d}x}{\rm artanh}(x)=\frac{1}{1-x^2}\ ,\ |x|<1$$

$$\frac{\mathrm{d}}{\mathrm{d}x}{\rm arsech}(x)=-\frac{1}{x\sqrt{1-x^2}}\ ,\ 01$$

$$\frac{\mathrm{d}}{\mathrm{d}x}{\rm arcsch}(x)=-\frac{1}{|x|\sqrt{1+x^2}}\ ,\ x\ne0$$