Question

What's the derivative of `arctanx^(1/2)-2x^(1/2)`?

Answer 1

`d/dx(arctansqrtx-2sqrtx) = -(2x+1)/(2(x+1)sqrtx)`

Explanation:

`d/dx arctanf(x) = (f'(x))/([f(x)]^2+1)`

`therefore d/dxarctansqrtx = (1/(2sqrtx))/(1+x) = 1/(2(x+1)sqrtx)`

`d/dx-2sqrtx = -2d/dx2sqrtx = -2* 1/(2sqrtx) = -1/sqrtx`

`therefore d/dx(arctansqrtx - 2sqrtx) = 1/(2(x+1)sqrtx)-1/sqrtx = 1/(2(x+1)sqrtx) -(2(x+1))/(2(x+1)sqrtx) = (1-2(x+1))/(2(x+1)sqrtx) = -(2x+1)/(2(x+1)sqrtx)`