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[parent] table of derivatives (Feature)

Below are some tables of some real-valued functions and their corresponding derivatives:

Basic rules

$ f(x)$ $ \displaystyle{\frac{df(x)}{dx}} = f'(x)$
$ f(x) + g(x)$ $ f'(x)+g'(x)$
$ f(x)g(x)$ $ f'(x)g(x)+f(x)g'(x)$
$ \displaystyle \frac{f(x)}{g(x)},\, g\neq 0$ $ \displaystyle \frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}$
$ f(g(x))$ $ f'(g(x))g'(x)$
$ f^{-1}(x)$ $ \displaystyle{\frac{1}{f'(f^{-1}(x))}}$


Polynomials and powers

$ f(x)$ $ f'(x)$ applicable domain
$ c\in \mathbb{R}$ 0 $ x\in \mathbb{R}$
$ x^r$ $ rx^{r-1}$ $ x\in \mathbb{R}$
$ \sqrt{x}$ $ \displaystyle\frac{1}{2\sqrt{x}}$ $ x>0$
$ \vert x\vert$ $ \displaystyle\frac{x}{\vert x\vert}=\frac{\vert x\vert}{x}$ $ x\ne 0$

Exponential and logarithmic functions

$ f(x)$ $ f'(x)$ applicable domain
% latex2html id marker 721 $ \exp(x)=e^x$ % latex2html id marker 723 $ \exp(x)=e^x$ $ x\in \mathbb{R}$
$ a^x$ $ a^x\ln{a}$ $ x\in \mathbb{R}$
$ \ln x$ $ \displaystyle{\frac{1}{x}}$ $ x>0$
$ x^x$ $ x^x(1+\ln x)$ $ x>0$


Trigonometric functions

$ f(x)$ $ f'(x)$ applicable domain
$ \sin{x}$ $ \cos{x}$ $ x\in \mathbb{R}$
$ \cos{x}$ $ -\sin{x}$ $ x\in \mathbb{R}$
$ \tan{x}$ $ \sec^2{x}$ $ x\ne n\pi+\displaystyle{\frac{\pi}{2}},\, n\in \mathbb{Z}$
$ \cot{x}$ $ -\csc^2{x}$ $ x\ne n\pi,\, n\in \mathbb{Z}$
$ \sec{x}$ $ \sec{x}\tan{x}$ $ x\ne n\pi+\displaystyle{\frac{\pi}{2}},\, n\in \mathbb{Z}$
$ \csc{x}$ $ -\csc{x}\cot{x}$ $ x\ne n\pi,\, n\in \mathbb{Z}$
$ \arcsin{x}$ $ \displaystyle\frac{1}{\sqrt{1-x^2}}$ $ \vert x\vert<1$
$ \arccos{x}$ $ \displaystyle-\frac{1}{\sqrt{1-x^2}}$ $ \vert x\vert<1$
$ \arctan{x}$ $ \displaystyle\frac{1}{1+x^2}$ $ x\in \mathbb{R}$


Hyperbolic functions

$ f(x)$ $ f'(x)$ applicable domain
$ \sinh{x}$ $ \cosh{x}$ $ x\in \mathbb{R}$
$ \cosh{x}$ $ \sinh{x}$ $ x\in \mathbb{R}$
$ \tanh{x}$ $ \operatorname{sech}^2{x}$ $ x\in \mathbb{R}$
$ \coth{x}$ $ -\operatorname{csch}^2{x}$ $ x\ne 0$
$ \operatorname{sech}{x}$ $ -\operatorname{sech}{x}\tanh{x}$ $ x\in \mathbb{R}$
$ \operatorname{csch}{x}$ $ -\operatorname{csch}{x}\coth{x}$ $ x\ne 0$
$ \operatorname{arsinh}{x}$ $ \displaystyle\frac{1}{\sqrt{x^2\!+\!1}}$ $ x\ne 0$
$ \operatorname{arcosh}{x}$ $ \displaystyle\frac{1}{\sqrt{x^2\!-\!1}}$ $ \vert x\vert>1$
$ \operatorname{artanh}{x}$ $ \displaystyle\frac{1}{1\!-\!x^2}$ $ -1 < x < 1$
$ \operatorname{arcoth}{x}$ $ \displaystyle\frac{1}{1\!-\!x^2}$ $ \vert x\vert > 1$

Other functions

(see error function, logarithmic integral, sine integral)
$ f(x)$ $ f'(x)$ applicable domain
Erf$ \,x\,$ $ \displaystyle\frac{2}{\sqrt{\pi}}e^{-x^2}$ $ x\in \mathbb{R}$
Li$ \,x$ $ \displaystyle\frac{1}{\ln{x}}$ $ x > 1$
Si$ \,x$ $ \displaystyle$sinc$ \,x$ $ x \in \mathbb{R}$

Instructions on how to add a function and its derivative. Open the entry in edit mode. Using the appropriate table for your function (or make a new table if applicable), make a copy of the two lines of comment (starting with %) in the code (within the tabular environment) and paste it immediately before the comment. For functions outside of the “Basic rules” section, include the appropriate domain. Uncomment the lines (take out the % symbols) after completing. Preview before saving the entry.



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See Also: table of integrals, derivative, general formulas for integration


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Cross-references: sine integral, logarithmic integral, error function, domain, derivatives, functions

This is version 24 of table of derivatives, born on 2007-10-12, modified 2008-05-15.
Object id is 9992, canonical name is TableOfDerivatives.
Accessed 2117 times total.

Classification:
AMS MSC: 26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
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