# Derivative Calculus

## Derivative Calculator

Input field for the function to be derived:

 clear Derivative $\frac{d}{dx}f\left(x\right)$ next Derivative $\frac{{d}^{n}}{d{x}^{n}}f\left(x\right)$ Plot Pos1 Ende 7 8 9 / $x$ 4 5 6 * ( ) 1 2 3 - a b c 0 . + sin cos tan ${e}^{x}$ $\mathrm{ln}\left(x\right)$ ${x}^{a}$ ^ asin acos atan ${x}^{2}$ $\sqrt{x}$ $\sqrt{x}$ $\sqrt{x}$ $\frac{\left(\right)}{\left(\right)}$ sinh cosh $\frac{ax+c}{bx+c}$ $\frac{a+x}{b+x}$ $\frac{{x}^{2}-{a}^{2}}{{x}^{2}+{a}^{2}}$ $\frac{1}{a+bx}$ $\frac{1+\sqrt{x}}{1-\sqrt{x}}$ $\sqrt{x+a}$ $\sqrt{{e}^{ax}}$ ${e}^{\sqrt{x}}$ $a{e}^{-b{x}^{2}+c}$ $\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$ $a{x}^{2}+bx+c$ ${e}^{x}\cdot \mathrm{sin}\left(x\right)\cdot \mathrm{cos}\left(x\right)$ $\frac{1}{\sqrt{ax}}$ $a{e}^{bx+c}$ ${e}^{ax}$ ${e}^{a{x}^{2}}$ $\frac{1}{{e}^{ax}}$ $\frac{x}{{e}^{x}}$ $\frac{1}{\mathrm{sin}}$ $\frac{1}{\mathrm{cos}}$ $\frac{1}{\mathrm{tan}}$ $a\cdot \mathrm{sin}\left(bx+c\right)$ $a\cdot \mathrm{cos}\left(bx+c\right)$ $a\cdot \mathrm{tan}\left(bx+c\right)$ $a\cdot {\mathrm{sin}}^{2}\left(bx+c\right)$

### Spellings

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

### Elementary derivatives

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

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

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

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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 differentiating
• Sum rule: When deriving a sum, the sum elements can be derived individually
• Product rule: Rule for deriving products
• Quotient Rule: Rule for deriving quotients
• Chain Rule: Nested functions go into a derivative product when differentiated

### Derivation rules in short

Factor rule: A constant factor is preserved when differentiating

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

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 differentiating. 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 differentiating.

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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 differentiating. 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.

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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.

## More Calculators

Here is a list of of further useful calculators:

Index Partial derivatives and gradient Gradient calculator Gradient 2d Plot Function Plot ODE first order