# Derivation of trigonometric functions

## Derivative Calculator

Function of x

First derivative of the function after x

Input field for the function:

$f\left(x\right)=$

 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 . + $\mathrm{sin}$ $\mathrm{cos}$ $\mathrm{tan}$ ${e}^{x}$ $\mathrm{ln}\left(x\right)$ ${x}^{a}$ ^ $\mathrm{asin}$ $\mathrm{acos}$ $\mathrm{atan}$ ${x}^{2}$ $\sqrt{x}$ $\sqrt{x}$ $\sqrt{x}$ $\frac{\left(\right)}{\left(\right)}$ $\mathrm{sinh}$ $\mathrm{cosh}$ $a\cdot \mathrm{sin}\left(bx+c\right)$ $a\cdot \mathrm{cos}\left(bx+c\right)$ $a\cdot \mathrm{tan}\left(bx+c\right)$ $\frac{1}{\mathrm{sin}\left(x\right)}$ $a\cdot {\mathrm{sin}}^{2}\left(bx+c\right)$ $a\cdot {\mathrm{cos}}^{2}\left(bx+c\right)$ $a\cdot {\mathrm{tan}}^{2}\left(bx+c\right)$ $\frac{1}{\mathrm{cos}\left(x\right)}$ $\frac{\mathrm{sin}\left(a\cdot x\right)}{\mathrm{cos}\left(b\cdot x\right)}$ ${e}^{x}\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)$ $\mathrm{sin}\left(\mathrm{cos}\left(x\right)\right)$ $\frac{1}{\mathrm{tan}\left(x\right)}$

## Notations

Notations for derivatives:

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

## Derivation 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)}$

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

$\frac{d}{dx}\mathrm{arcsin}\left(x\right)=\frac{1}{\sqrt{1-{x}^{2}}}$

$\frac{d}{dx}\mathrm{arccos}\left(x\right)=-\frac{1}{\sqrt{1-{x}^{2}}}$

$\frac{d}{dx}\mathrm{arctan}\left(x\right)=\frac{1}{1+{x}^{2}}$

$\frac{d}{dx}\mathrm{arccot}\left(x\right)=-\frac{1}{1+{x}^{2}}$

### 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

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

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