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Derivative

Tool for calculating function derivatives (simple derivative or partial derivative). Formal calculator from an expression f(x) of the function to be differentiated.

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Derivative -

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# Derivative

## Partial Derivative Calculator

Use the derivative calculator above by indicating a single variable (that of the partial derivative).

## Primitive Calculator

### What is a derivative? (Definition)

The differentiation is a fundamental tool when analyzing a function, it allows to measure the sensitivity to change of a function.

In other words, the derivative of a function is the measure of the change in the function at a given point, it tells in what proportions the function is changing at that point.

The derivative of a function $f$ is denoted $f'$ (with an apostrophe named prime) or $\frac{d}{dx}f$ where $d$ is the derivative operator and $x$ the variable on which to derivate.

### How to calculate a derivative?

The derivative (or first derivative) calculation applies the general formula $$\frac{d}{dx}f = f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$

In practice, this limit calculation is sometimes laborious, it is easier to learn the list of usual derivatives, already calculated and known (see below).

On dCode, the derivative calculator knows all the derivatives, indicate the function and the variables on which to derivate/differentiate in order to obtain the result of the derivative computation.

Example: $$f(x) = x^2+\sin(x) \Rightarrow f'(x) = 2 x+\cos(x)$$

Derivative calculus is often used in physics to compute a velocity.

### How to calculate a partial derivative?

A partial derivative is a derivative that only applies to one variable, leaving the others intact (considering them as constants).

On dCode, indicate a single variable if the function has several to obtain a partial derivative.

To obtain all the partial derivatives, perform several calculations, with the same function, but changing the letter of the variable.

### How to calculate a cross derivative?

A cross derivative is a partial derivative with respect to 2 variables, while leaving any other variables unchanged/constant.

On dCode, indicate the 2 variables one after the other to obtain the result of a cross derivative.

### What is the list of common derivatives?

The more useful derivatives are:

NameFunctionDerivative
constant/number$$k \in \mathbb{R}$$$$0$$
variable$$x$$$$1$$
power n (exponent)$$x^n$$$$n x^{n-1}$$
negative power$$x^{-n}$$$$-n x^{-n-1}$$
inverse$$\frac{1}{x}$$$$-\frac{1}{x^2}$$
inverse power$$\frac{1}{x^n}$$$$-\frac{n}{x^{n+1}}$$
root$$\sqrt{x}$$$$\frac {1}{2\sqrt{x}}$$
nth root$$\sqrt[n]x$$$$\frac{1}{n\sqrt[n]{x^{n-1}}}$$
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 x$$a^x$$$$a^x \ln a$$
sine$$\sin(x)$$$$\cos(x)$$
cosine$$\cos(x)$$$$- \sin(x)$$
tangent$$\tan(x)$$$$\frac{1}{\cos^2(x)} \\ = \sec^2(x) \\ = 1+\tan^2(x) \\ = \frac{2}{1+\cos(2x)}$$
secant$$\sec(x) = \frac{1}{\cos(x)}$$$$\frac{\tan(x)}{\cos(x)} \\ = \sec(x)\tan(x) \\ = \frac{2\sin(x)}{1+\cos(2x)}$$
cosecant$$\csc(x) = \frac{1}{\sin(x)}$$$$-\frac{\cos(x)}{\sin^2(x)} \\ = -\cot(x)\csc(x) \\ = \frac{2\cos(x)}{-1+\cos(2x)}$$
cotangent$$\cot(x) = \frac{1}{\tan(x)}$$$$- \frac{1}{\sin^2(x)} \\ = -1-\cot^2(x) \\ = -\csc^2(x) \\ = \frac{2}{-1+\cos(2x)}$$
arcsine$$\arcsin(x)$$$$\frac{1}{\sqrt{1-x^2}}$$
arccosine$$\arccos(x)$$$$-\frac{1}{\sqrt{1-x^2}}$$
arctangent$$\arctan(x)$$$$\frac{1}{1+x^2}$$
sine hyperbolic$$\sinh(x)$$$$\cosh(x)$$
cosine hyperbolic$$\cosh(x)$$$$\sinh(x)$$
tangent hyperbolic$$\tanh(x)$$$$\frac{1}{\cosh^2(x)} \\ = 1 - \tanh^2(x)$$
cotangent hyperbolic$$\coth(x)$$$$\frac{-1}{\sinh^2(x)} \\ = 1 - \coth^2(x)$$
arcsinus hyperbolic$$\operatorname{arcsinh}(x)$$$$\frac{1}{\sqrt{1+x^2}}$$
arccosinus hyperbolic$$\operatorname{arccosh}(x)$$$$\frac{1}{\sqrt{x^2-1}}$$
arctangent hyperbolic$$\operatorname{arctanh}(x)$$$$\frac{1}{1-x^2}$$

### What is the list of compound function derivatives?

Common derivatives of compound functions to know are:

NameCompound FunctionDerivative
compound function$$g \circ f$$$$(g' \circ f)\times f'$$
function power n (exponentiation)$$f^n$$$$n f^{n - 1} f'$$
sine of function$$\sin(f)$$$$f' \cos(f)$$
cosine of function$$\cos(f)$$$$- f' \sin(f)$$
exponential of function$$\exp(f)$$$$f' \exp(f)$$
square root of function
(positive function)
$$\sqrt{f}$$$$\frac{f'}{2\sqrt{f}}$$
logarithm of function
(positive function)
$$\ln(f)$$$$\frac{f'}{f}$$

### How to calculate a second derivative?

A second derivative is equivalent to compute the derivative twice, for dCode, indicate twice the same variable to get the second derivative.

Second derivative calculations are often used in physics to compute acceleration (derivative of velocity).

### What is the difference between derivative and primitive?

The derivative calculation is the inverse operation of the primitive/antiderivative calculation (indefinite integral).

dCode has a tool for calculating primitives.

### What is a derivator?

A derivator is a mathematical operator that has nothing to do with the derivative operation.

It is nevertheless possible, by extension, to call the dCode tool on this page an online derivator allowing to calculate derivatives.

The less common term derivation can be use to describe the process of finding a derivative.

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Derivative on dCode.fr [online website], retrieved on 2024-09-13, https://www.dcode.fr/derivative

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