# Derivative Calculator and Derivative Calculus

## Derivatives

Derivatives of basic functions.

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

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

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

#### Derivative of trigonometric functions

$\frac{d}{dx}\mathrm{sin}\left(x\right)=\mathrm{cos}\left(x\right)$

$\frac{d}{dx}\mathrm{cos}\left(x\right)=-\mathrm{sin}\left(x\right)$

$\frac{d}{dx}\mathrm{sin}\left(kx\right)=k\mathrm{cos}\left(kx\right)$

$\frac{d}{dx}\mathrm{cos}\left(kx\right)=-k\mathrm{sin}\left(kx\right)$

$\frac{d}{dx}\mathrm{tan}\left(x\right)=\frac{d}{dx}\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}=\frac{1}{{\mathrm{cos}}^{2}\left(x\right)}$

#### Derivation exponential and logarithmic functions

$\frac{d}{dx}{e}^{x}={e}^{x}$

$\frac{d}{dx}\mathrm{ln}\left(x\right)=\frac{1}{x}$

$\frac{d}{dx}{\mathrm{log}}_{a}\left(x\right)=\frac{1}{x}{\mathrm{log}}_{a}\left(e\right)$

## Derivative Calculator

The derivative calculator calculates the first and second derivative of a function f(x).

f(x) =
 cl d/dx Pos1 End 7 8 9 / x 4 5 6 * ( ) 1 2 3 - a b c 0 . + sin cos tan ex ln ^ asin acos atan x2 √x xa sinh cosh a⋅sin(b⋅x+c) a⋅e(-b⋅x2+c) a⋅x2+b⋅x+c e(x)⋅sin(x)⋅cos(x)

## Differential Calculus

### Derivative calculus rules

The following are the most important differentiation rules described and illustrated by examples.

• Factor Rule
• Sum rule
• Product Rule
• Quotient Rule
• Chain Rule

#### Factor rule and Sum rule

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

$\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)$

Derivation of the summands

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

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

The constant factor a is retained when deriving

#### Example of factor and sum rule

$f\left(x\right)$$=2{x}^{2}+\frac{2}{3}{x}^{3}$

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

$\frac{d}{dx}\left(2{x}^{2}+\frac{2}{3}{x}^{3}\right)$$=\frac{d}{dx}2{x}^{2}+\frac{d}{dx}\frac{2}{3}{x}^{3}$

Application of the sum rule.

$\frac{d}{dx}2{x}^{2}+\frac{d}{dx}\frac{2}{3}{x}^{3}$$=2\frac{d}{dx}{x}^{2}+\frac{2}{3}\frac{d}{dx}{x}^{3}$

Apply the factor rule in each part of the sum.

$2\frac{d}{dx}{x}^{2}+\frac{2}{3}\frac{d}{dx}{x}^{3}$$=4x+2{x}^{2}$

Deriving the terms gives the derivation of the example function f.

#### Product rule

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.

#### Examples for Product rule

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.

#### Quotient rule

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

#### Chain rule

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 considered as variable. That it is not differentiated by x but by the inner function g.

#### Mixed differentiations

The following examples show the mixed use of the derivation rules.