Question

Differentiate the following w.r.t. x:

`sqrt(e^(sqrtx)), x > 0`

Answer 1

Let, y = `sqrt(e^(sqrtx)), x > 0`

Differentiating both sides with respect to x,

`dy/dx = d/dx (e^(sqrtx))^(1/2)`

or `dy/dx = 1/2 (e^(sqrtx))^(1/2 - 1) d/dx e^(sqrtx)`

`= 1/2 (e^(sqrtx))^(-1//2) e^(sqrtx) d/dx x^(1/2)`

`= 1/2 (e^(sqrtx))^(-1//2) e^(sqrtx) 1/2 x^(-1 //2) = 1/4 1/(sqrt(e^(sqrtx))) e^(sqrtx) 1/sqrtx`

`= 1/4 e^(sqrtx)/(sqrt(x. e^(sqrtx))), x > 0`