# What is the derivative of this function tan^-1(5x)?

Aug 20, 2016

$\frac{5}{1 + 25 {x}^{2}}$

#### Explanation:

differentiate using the $\textcolor{b l u e}{\text{chain rule}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \times \frac{\mathrm{du}}{\mathrm{dx}}} \textcolor{w h i t e}{\frac{a}{a}} |}}} \ldots \ldots . . \left(A\right)$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder }} \textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\frac{d}{\mathrm{dx}} \left({\tan}^{-} 1 x\right) = \frac{1}{1 + {x}^{2}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

let $u = 5 x \Rightarrow \frac{\mathrm{du}}{\mathrm{dx}} = 5$

and y=tan^-1 urArr(dy)/(du)=1/(1+u^2

Substitute these values into (A) and convert u back into x.

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{1 + {\left(5 x\right)}^{2}} .5 = \frac{5}{1 + 25 {x}^{2}}$