# Derivative Calculus

## Derivative Calculator

Input field for the function to be derived:

f(x) =

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Derivative $\frac{d}{dx}f\left(x\right)$
next Derivative $\frac{{d}^{n}}{d{x}^{n}}f\left(x\right)$
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$\mathrm{sin}$
$\mathrm{cos}$
$\mathrm{tan}$
${e}^{x}$
$\mathrm{ln}\left(x\right)$
${x}^{a}$
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$\mathrm{asin}$
$\mathrm{acos}$
$\mathrm{atan}$
${x}^{2}$
$\sqrt{x}$
$\sqrt[3]{x}$
$\sqrt[4]{x}$
$\frac{\left(\right)}{\left(\right)}$
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$\mathrm{sinh}$
$\mathrm{cosh}$
$\frac{ax+c}{bx+c}$
$\frac{a+x}{b+x}$
$\frac{{x}^{2}-{a}^{2}}{{x}^{2}+{a}^{2}}$
$\frac{1}{a+bx}$
$\frac{1+\sqrt{x}}{1-\sqrt{x}}$
$\sqrt{x+a}$
$\sqrt{{e}^{ax}}$
${e}^{\sqrt{x}}$
$\frac{1}{\sqrt{ax}}$
$\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$
${e}^{x}\cdot \mathrm{sin}\left(x\right)\cdot \mathrm{cos}\left(x\right)$
$\frac{1}{\mathrm{sin}}$
$\frac{1}{\mathrm{cos}}$
$\frac{1}{\mathrm{tan}}$
${e}^{\sqrt{x}}$
$a{e}^{-b{x}^{2}+c}$
$\sqrt{{e}^{ax}}$
$a{e}^{bx+c}$
${e}^{a{x}^{2}}$
$\frac{1}{{e}^{ax}}$
$\frac{x}{{e}^{x}}$
$a\cdot \mathrm{sin}\left(bx+c\right)$
$a\cdot \mathrm{cos}\left(bx+c\right)$
$a\cdot \mathrm{tan}\left(bx+c\right)$
$a\cdot {\mathrm{sin}}^{2}\left(bx+c\right)$
$a{x}^{2}+bx+c$
FunctionDescription
sin(x)Sine of x
cos(x)Cosine of x
tan(x)Tangent of x
asin(x)arcsine
acos(x)arccosine of x
atan(x)arctangent of x
atan2(y, x)Returns the arctangent of the quotient of its arguments.
cosh(x)Hyperbolic cosine of x
sinh(x)Hyperbolic sine of x
pow(a, b)Power ab
sqrt(x)Square root of x
exp(x)e-function
log(x), ln(x)Natural logarithm
log(x, b)Logarithm to base b
log2(x), lb(x)Logarithm to base 2
log10(x), ld(x)Logarithm to base 10
more ...

## Notation for Differentiation

For differentiation there are different notations usual with the same meaning. The usefulness of each notation varies with the context. The most common notations for differentiation from Leibnitz, Euler, Lagrange and Newton are listed below.

### Leibnitz Notation for Differentiation

The derivative in Leibnitz notation of a function f to the variable x is given as follows.

$\frac{d}{dx}f\left(x\right)=\frac{df}{dx}\left(x\right)=\frac{df\left(x\right)}{dx}$

Usual is also the setting y = f(x) with the notation for the derivative as follows.

$\frac{dy}{dx}$

Second, third and higher derivatives are written as follows.

$\frac{{d}^{2}y}{d{x}^{2}};\frac{{d}^{3}y}{d{x}^{3}};...;\frac{{d}^{n}y}{d{x}^{n}};$

### Lagrange Notation for Differentiation

The first derivative in Lagrange notation is given by a ' at the function.

${f}^{\prime }\left(x\right)$

The higher derivatives in Lagrange notation are given as follows.

${f}^{″}\left(x\right);{f}^{‴}\left(x\right);{f}^{\left(4\right)}\left(x\right);...;{f}^{\left(n\right)}\left(x\right)$

### Euler Notation for Differentiation

Euler uses the D operator for the derivative.

$Df=\frac{d}{dx}f\left(x\right)$

### Newton Notation for Differentiation

Newton's notation is also called dot notation. The notation uses dots to notated the derivatives. This notation is used for functions depending on time t.

$\stackrel{˙}{f}\left(t\right)=\frac{df}{dt}$

The higher derivatives in Newton notation are given as follows.

$\stackrel{¨}{f}\left(t\right)=\frac{{d}^{2}f}{d{t}^{2}};\stackrel{⃛}{f}\left(t\right)=\frac{{d}^{3}f}{d{t}^{3}}$

## Basic derivatives

$\frac{d}{dx}\mathrm{Const.}=0$

$\frac{d}{dx}x=1$

$\frac{d}{dx}{x}^{n}=n\cdot {x}^{n-1}$

$\frac{d}{dx}\frac{1}{x}=-\frac{1}{{x}^{2}}$

$\frac{d}{dx}\frac{1}{{x}^{n}}=-\frac{n}{{x}^{n+1}}$

$\frac{d}{dx}{a}^{x}={a}^{x}\mathrm{ln}a$

$\frac{d}{dx}{a}^{kx}={a}^{kx}k\mathrm{ln}a$

$\frac{d}{dx}\frac{a\cdot x+c}{b\cdot x+c}=\frac{c\cdot \left(a-b\right)}{{\left(b\cdot x+c\right)}^{2}}$

$\frac{d}{dx}\frac{1}{a+b\cdot x}=\frac{-b}{{\left(a+b\cdot x\right)}^{2}}$

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## Derivation Rules

In the following, the most important derivation rules are described and explained using examples.

• Factor rule: A constant factor is retained when differentiate
• Sum rule: Rule for the derivation of a sum
• Product rule: Rule for the derivation of products
• Quotient Rule: Rule for the derivation of quotients
• Chain Rule: Rule for the derivation of nested functions

### Derivation rules in short

Factor rule: A constant factor is preserved when differentiate

$\left(a\cdot f\right)\prime =a\cdot f\prime$

Sum rule: When deriving a sum, the summands can be derived individually

$\left({f}_{1}+{f}_{2}\right)\prime ={f}_{1}\prime +{f}_{2}\prime$

Product rule: Rule for deriving products

$\left(u\cdot v\right)\prime =u\prime \cdot v+u\cdot v\prime$

Quotient rule: Rule for deriving quotients

$\left(\frac{u}{v}\right)\prime =\frac{u\prime \cdot v-u\cdot v\prime }{{v}^{2}}$

Chain rule: Nested functions go into a product of the inner and outer derivatives when differentiated

$\left(f\left(g\left(x\right)\right)\right)\prime =f\prime \left(g\right)\cdot g\prime \left(x\right)$

### Factor and Sum Rule

$\frac{d}{dx}f\left(x\right)=\frac{d}{dx}\left({a}_{1}{f}_{1}\left(x\right)+{a}_{2}{f}_{2}\left(x\right)\right)=\frac{d}{dx}{a}_{1}{f}_{1}\left(x\right)+\frac{d}{dx}{a}_{2}{f}_{2}\left(x\right)={a}_{1}\frac{d}{dx}{f}_{1}\left(x\right)+{a}_{2}\frac{d}{dx}{f}_{2}\left(x\right)$

The sum rule states that the sum elements can be individually differentiated.

Derivation of the summands

$\frac{d}{dx}\left({f}_{1}\left(x\right)+{f}_{2}\left(x\right)\right)=\frac{d}{dx}{f}_{1}\left(x\right)+\frac{d}{dx}{f}_{2}\left(x\right)$

The factor rule states that the constant factors are conserved during derivation.

The constant factor a is retained when deriving

$\frac{d}{dx}\left(af\left(x\right)\right)=a\frac{d}{dx}f\left(x\right)$

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#### Example of the application of the factor and sum rule (open by selection)

The example function contains sum and constant factors. To differentiate, both rules are applied.

In the first step, the sum rule is applied. In the second step, the factor rule on each addend and finally derives the individual terms, the derivative of the function.

### Product Rule

$\frac{d}{dx}f\left(x\right)=\frac{d}{dx}\left(u\left(x\right)\cdot v\left(x\right)\right)=v\left(x\right)\frac{d}{dx}u\left(x\right)+u\left(x\right)\frac{d}{dx}v\left(x\right)={u}^{\prime }\left(x\right)\cdot v\left(x\right)+u\left(x\right)\cdot {v}^{\prime }\left(x\right)$

The product rule specifies how to handle the product of two functions when differentiate. In words, the product rule can be expressed as follows: Derivation of the first function times the second function plus the first function times derivation of the second function.

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#### Examples of the application of the product rule (open by selection)

Here are some examples of applying the product rule.

#### Product rule Example 1

In the first example, the product rule is explained using a function consisting of the product of the sine and cosine functions. The derivative is done according to the product rule so that the derivative of the first factor is multiplied by the second factor and added to the derivative of the second factor multiplied by the first factor.

#### Product rule Example 2

In the second example, the product rule is explained by a function consisting of the product of the exponential and sine functions.

Derivation takes place according to the product rule as in the first example only that the first factor here is the e-function and the second is the sine function.

#### Product rule Example 3

In the third example, the product rule is explained by a function consisting of the product of three functions.

If a product consists of more than two functions, then the product rule can be used successively by combining functions as required and applying the product rule several times in succession.

By the respective parenthesis one gets again a product of two factors on which one can apply the product rule. Here in the example we continue with the first variant.

### Quotient Rule

$\frac{d}{dx}f\left(x\right)=\frac{d}{dx}\frac{u\left(x\right)}{v\left(x\right)}=\frac{v\left(x\right)\frac{d}{dx}u\left(x\right)-u\left(x\right)\frac{d}{dx}v\left(x\right)}{{v}^{2}\left(x\right)}=\frac{{u}^{\prime }\left(x\right)\cdot v\left(x\right)-u\left(x\right)\cdot {v}^{\prime }\left(x\right)}{{v}^{{}^{2}}}$

The quotient rule specifies how to treat the quotient of two functions when differentiate.

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#### Example of the application of the quotient rule (open by selection)

As an example for the application of the quotient rule, the quotient of the sine and the cosine function is used. The application is similar to the product rule. The role of the factors takes here each numerator and denominator of the break.

### Chain Rule

$\frac{d}{dx}f\left(g\left(x\right)\right)=\frac{d}{dx}g\left(x\right)\cdot \frac{d}{dg}f\left(g\right)={g}^{\prime }\left(x\right)\cdot {f}^{\prime }\left(g\right)$

The chain rule specifies how nested functions are to be treated when differentiate. One distinguishes between the inner function and the outer function. Thus, the chain rule can be formulated as follows: the derivative is derivative of the inner function times the derivative of the outer function. In the derivation of the outer function, the inner function as a whole is regarded as variable. That it is not differentiated by x but by the inner function g.

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#### Examples of the application of the chain rule (open by selection)

Here are some examples of applying the chain rule. In the first example, the sine function is in the exponent of the e-function. The sine function is therefore the inner function g. The second example shows how to differentiate a power function. In the third example, a quadratic function is within a trigonometric function.

### Mixed use

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#### Examples of the mixed use of derivation rules (open by selection)

Here are some examples of the mixed use of derivation rules. The first example uses product and quotient rules. The second example shows how product and chain rule can be used. The third example uses sum, factor and chain rules.

## Differentiate Vectors

Vectors will be differentiate by derivation all vector components.

$\frac{d}{dt}\stackrel{\to }{f\left(t\right)}=\left(\begin{array}{c}\frac{d}{dt}{f}_{1}\left(t\right)\\ \frac{d}{dt}{f}_{2}\left(t\right)\\ ⁝\\ \frac{d}{dt}{f}_{n}\left(t\right)\end{array}\right)$

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#### Example of differentiating a vector function

In the following example, the derivative of a vector function is given using the parameter representation of a 3-dimensional curve.

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#### Rules for differentiating vector functions

In the following some rules for the differentiation of vector functions are given. Among them also the derivation of cross product and scalar product of vector functions. f denotes a scalar function. With the cross product the factors may not be exchanged.

## Partial Derivatives

For functions with more than one variables the derivative to one of the variables is called partial derivative.

For a function with the variable x and several further variables the partial derivative to x is noted as follows.

$\frac{\mathrm{\partial }}{\mathrm{\partial }x}f\left(x,y,...\right)$

For partial derivation, the other variables are treated as constants.

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#### Example for partial derivatives

In the following example, the derivative of a function of x, y, and z is derived partially according to each of the variables.

A gradient is a vector whose components are the partial derivatives of a function f. There are two common names for the gradient. One is grad(f) and the other uses the differential operator nabla ∇.

$grad\left(f\right)=\mathrm{\nabla }f=\left(\begin{array}{c}\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_{1}}\\ \frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_{2}}\\ ⁝\\ \frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_{n}}\end{array}\right)$

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The following calculation rules apply to the gradient.

## Implicit derivation

A function F(x, f(x)) = 0 can also be differentiated without explicitly solving the function if the corresponding partial derivatives exist.

If we set y = f(x) and thus F(x, y) = 0 for a clearer notation, then the derivative can be calculated by means of partial derivatives as follows.

$F\left(x,f\left(x\right)\right)=F\left(x,y\right)=0$

${f}^{\prime }\left(x\right)=\frac{dy}{dx}=-\frac{\frac{\mathrm{\partial }}{\mathrm{\partial }x}F\left(x,y\right)}{\frac{\mathrm{\partial }}{\mathrm{\partial }y}F\left(x,y\right)}$

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#### Example of implicit derivation

Example of the derivation of an implicit function.