Question

What is the derivative of `arctan sqrt x`?

Answer 1

You can do it two ways. If you remember the actual derivative:

`d/(dx)[arctanu] = 1/(1+u^2)((du)/(dx))`

`d/(dx)[arctansqrtx] = 1/(1+x)*1/(2sqrtx) = color(blue)(1/(2sqrtx(1+x)))`

Or, you can implicitly derive it.

`y = arctansqrtx`

`tany = sqrtx`

`sec^2y((dy)/(dx)) = sec^2(arctansqrtx) = 1/(2sqrtx)`

`(1 + tan^2(arctansqrtx))((dy)/(dx)) = 1/(2sqrtx)`

`(1 + x)((dy)/(dx)) = 1/(2sqrtx)`

`color(blue)((dy)/(dx) = 1/(2sqrtx(1 + x))`