Question

How do you differentiate `y=tan^-1(5x)`?

Answer 1

`dy/dx=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)`

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

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

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

substitute results into (A) and change u back into terms of x.

`rArrdy/dx=1/(1+u^2)xx5=5/(1+25x^2)`