Question

What's the derivative of `arctan (x/2)`?

Answer 1

`2/(4+x^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)`

Note that `x/2=1/2x`

let `color(blue)(u=1/2x)rArr(du)/(dx)=1/2`

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

and `y=arctancolor(blue)(u)rArr(dy)/(du)=1/(1+color(blue)(u)^2`

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

`dy/dx=1/(1+(x/2)^2)xx1/2=(1/2)/((1+x^2/4))=(1/2)/(1/4(4+x^2))`

`rArrdy/dx=2/(4+x^2)`