Question

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

Answer 1

`5/(1+25x^2)`

Explanation:

differentiate using the `color(blue)"chain rule"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(dy/dx=(dy)/(du)xx(du)/(dx))color(white)(a/a)|)))........ (A)`

`color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(d/dx(tan^-1 x)=1/(1+x^2))color(white)(a/a)|)))`

let `u=5xrArr(du)/(dx)=5`

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

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

`rArrdy/dx=1/(1+(5x)^2) .5=5/(1+25x^2)`