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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

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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:

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.

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