Question

How do you find the derivative of the function: `arctan (cos x)`?

Answer 1

`(-sinx)/(1+cos^2x)`

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(arctanx)=1/(1+x^2))color(white)(a/a)|)))`

let `u=cosxrArr(du)/(dx)=-sinx`

and `y=arctanurArr(dy)/(du)=1/(1+u^2)`

substitute these values into (A) changing u back to x.

`rArrdy/dx=1/(1+u^2)xx(-sinx)=(-sinx)/(1+cos^2x)`