Tool to compute derivatives. The differentiation is a fundamental tool when analyzing a function, it allows to measure the sensitivity to change of a function.
Derivative - dCode
Tag(s) : Mathematics
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Derivative
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Derivative Calculator
Tool to compute derivatives. The differentiation is a fundamental tool when analyzing a function, it allows to measure the sensitivity to change of a function.
Answers to Questions
How to calculate a derivative?
Mathematicians have defined derivatives using the formula $$ \frac{d}{dx}f = f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} $$ The derivative calculation is the inverse operation of primitive calculation. dCode knows all derivatives, just indicate the function and the deriving variable.
Example: $$ f(x) = x^2+\sin(x) \Rightarrow f'(x) = 2 x+\cos(x) $$
What is the list of common derivatives?
The more useful derivatives are:
| Name | Function | Derivative |
|---|---|---|
| constant | $$ k, in, \mathbb{R} $$ | $$ 0 $$ |
| variable | $$ x $$ | $$ 1 $$ |
| power n | $$ x^n $$ | $$ n x^{n-1} $$ |
| negative power | $$ x^{-n} $$ | $$ -n x^{-n-1} $$ |
| fraction | $$ \frac{1}{x} $$ | $$ -\frac{1}{x^2} $$ |
| inverse power | $$ \frac1{x^n} $$ | $$ -\frac n{x^{n+1}} $$ |
| root | $$ \sqrt{x} $$ | $$ \frac 1{2\sqrt{x}} $$ |
| nth root | $$ \sqrt[n]x $$ | $$ \frac1{n\sqrt[n]{x^{n-1}}} $$ |
| fractional power | $$ x^{1/n} $$ | $$ (1/n)x^{(1/n)-1} $$ |
| natural logarithm | $$ \ln |x| $$ | $$ \frac{1}{x} $$ |
| logarithm base a | $$ \log_a |x| $$ | $$ \frac{1}{x \ln a} $$ |
| exponential | $$ e^x $$ | $$ e^x $$ |
| exponent | $$ a^x $$ | $$ a^x \ln a $$ |
| sinus | $$ \sin x $$ | $$ \cos x $$ |
| cosinus | $$ \cos x $$ | $$ - \sin x $$ |
| tangent | $$ \tan x $$ | $$ \frac{1}{\cos^2 x} = 1+\tan^2 x $$ |
| cotangent | $$ \cot x $$ | $$ - \frac{1}{\sin^2 x} = -1-\cot^2 x $$ |
| arcsinus | $$ \arcsin x $$ | $$ \frac{1}{\sqrt{1-x^2}} $$ |
| arccosinus | $$ \arccos x $$ | $$ -\frac{1}{\sqrt{1-x^2}} $$ |
| arctangent | $$ \arctan x $$ | $$ \frac{1}{1+x^2} $$ |
| sinus hyperbolic | $$ \sinh x $$ | $$ \cosh x $$ |
| cosinus hyperbolic | $$ \cosh x $$ | $$ \sinh x $$ |
| tangent hyperbolic | $$ \tanh x $$ | $$ \frac{1}{\cosh^2 x} = 1 - \tanh^2 x $$ |
| cotangent | $$ \coth $$ | $$ \frac{-1}{\sinh^2 x} = 1 - \coth^2 x $$ |
| arcsinus hyperbolic | $$ \mathrm{arcsinh} x $$ | $$ \frac{1}{\sqrt{1+x^2}} $$ |
| arccosinus hyperbolic | $$ \mathrm{arccosh} x $$ | $$ \frac{1}{\sqrt{x^2-1}} $$ |
| arctangent hyperbolic | $$ \mathrm{arctanh} x $$ | $$ \frac{1}{1-x^2} $$ |
How to calculate a second derivative?
It is equivalent to compute the derivative twice, for dCode, indicate twice the same variable.
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Source code
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