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Derivative

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.

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Derivative -

Tag(s) : Mathematics

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# Derivative

## 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.

### 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:

NameFunctionDerivative
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.