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table of derivatives
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(Feature)
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Below are some tables of some real-valued functions and their corresponding derivatives:
| $f(x)$ |
$\displaystyle{\frac{df(x)}{dx}} = f'(x)$ |
| $f(x) + g(x)$ |
$f'(x)+g'(x)$ |
| $f(x)g(x)$ |
$f'(x)g(x)+f(x)g'(x)$ |
| $\displaystyle \frac{f(x)}{g(x)},\, g\neq 0$ |
$\displaystyle \frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}$ |
| $f(g(x))$ |
$f'(g(x))g'(x)$ |
| $f^{-1}(x)$ |
$\displaystyle{\frac{1}{f'(f^{-1}(x))}}$ |
| $f(x)$ |
$f'(x)$ |
applicable domain |
| $c\in \mathbb{R}$ |
0 |
$x\in \mathbb{R}$ |
| $x^r$ |
$rx^{r-1}$ |
$x\in \mathbb{R}$ |
| $\sqrt{x}$ |
$\displaystyle\frac{1}{2\sqrt{x}}$ |
$x>0$ |
| $|x|$ |
$\displaystyle\frac{x}{|x|}=\frac{|x|}{x}$ |
$x\ne 0$ |
| $f(x)$ |
$f'(x)$ |
applicable domain |
| $\exp(x)=e^x$ |
$\exp(x)=e^x$ |
$x\in \mathbb{R}$ |
| $a^x$ |
$a^x\ln{a}$ |
$x\in \mathbb{R}$ |
| $\ln x$ |
$\displaystyle{\frac{1}{x}}$ |
$x>0$ |
| $x^x$ |
$x^x(1+\ln x)$ |
$x>0$ |
| $f(x)$ |
$f'(x)$ |
applicable domain |
| $\sin{x}$ |
$\cos{x}$ |
$x\in \mathbb{R}$ |
| $\cos{x}$ |
$-\sin{x}$ |
$x\in \mathbb{R}$ |
| $\tan{x}$ |
$\sec^2{x}$ |
$x\ne n\pi+\displaystyle{\frac{\pi}{2}},\, n\in \mathbb{Z}$ |
| $\cot{x}$ |
$-\csc^2{x}$ |
$x\ne n\pi,\, n\in \mathbb{Z}$ |
| $\sec{x}$ |
$\sec{x}\tan{x}$ |
$x\ne n\pi+\displaystyle{\frac{\pi}{2}},\, n\in \mathbb{Z}$ |
| $\csc{x}$ |
$-\csc{x}\cot{x}$ |
$x\ne n\pi,\, n\in \mathbb{Z}$ |
| $\arcsin{x}$ |
$\displaystyle\frac{1}{\sqrt{1-x^2}}$ |
$|x|<1$ |
| $\arccos{x}$ |
$\displaystyle-\frac{1}{\sqrt{1-x^2}}$ |
$|x|<1$ |
| $\arctan{x}$ |
$\displaystyle\frac{1}{1+x^2}$ |
$x\in \mathbb{R}$ |
| $f(x)$ |
$f'(x)$ |
applicable domain |
| $\sinh{x}$ |
$\cosh{x}$ |
$x\in \mathbb{R}$ |
| $\cosh{x}$ |
$\sinh{x}$ |
$x\in \mathbb{R}$ |
| $\tanh{x}$ |
$\sech^2{x}$ |
$x\in \mathbb{R}$ |
| $\coth{x}$ |
$-\csch^2{x}$ |
$x\ne 0$ |
| $\sech{x}$ |
$-\sech{x}\tanh{x}$ |
$x\in \mathbb{R}$ |
| $\csch{x}$ |
$-\csch{x}\coth{x}$ |
$x\ne 0$ |
| $\arsinh{x}$ |
$\displaystyle\frac{1}{\sqrt{x^2\!+\!1}}$ |
$x\ne 0$ |
| $\arcosh{x}$ |
$\displaystyle\frac{1}{\sqrt{x^2\!-\!1}}$ |
$|x|>1$ |
| $\artanh{x}$ |
$\displaystyle\frac{1}{1\!-\!x^2}$ |
$-1 < x < 1$ |
| $\arcoth{x}$ |
$\displaystyle\frac{1}{1\!-\!x^2}$ |
$|x| > 1$ |
(see error function, logarithmic integral, sine integral)
| $f(x)$ |
$f'(x)$ |
applicable domain |
| $\mbox{Erf}\,x\,$ |
$\displaystyle\frac{2}{\sqrt{\pi}}e^{-x^2}$ |
$x\in \mathbb{R}$ |
| $\mbox{Li}\,x$ |
$\displaystyle\frac{1}{\ln{x}}$ |
$x > 1$ |
| $\mbox{Si}\,x$ |
$\displaystyle\mbox{sinc}\,x$ |
$x \in \mathbb{R}$ |
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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"table of derivatives" is owned by CWoo. [ full author list (4) ]
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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 5929 times total.
Classification:
| AMS MSC: | 26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems) |
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Pending Errata and Addenda
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