Question

What is the derivative of this function `(cos(e^(x^5*sin(x))))^(1/2)`?

Answer 1

`(-x^4e^(x^5sinx)sin(e^(x^5sinx))(xcosx + 5sinx))/(2sqrt(cos(e^(x^5sinx)))`

Explanation:

Given: `y = (cos(e^(x^5sinx)))^(1/2)`

Use the chain rule to differentiate.

Derivative rules needed:

Power rule: `" "(u^n)' = n u^(n-1)`

`(cos u)' = -u' sin u; " "(e^u)' = u' e^u`

product rule: `" "(uv)' = uv' + v u'`

Start out with the power rule where `u = cos(e^(x^5sinx)); n = 1/2`:

`y' = 1/2(cos(e^(x^5sinx)))^(-1/2) * (du)/(dx)(cos u)`

Let `u = e^(x^5sinx)`:

`y' = 1/2(cos(e^(x^5sinx)))^(-1/2) (-sin(e^(x^5 sinx)))* (du)/(dx)(e^u)`

Let `u = x^5sinx`:

`y' = 1/2(cos(e^(x^5sinx)))^(-1/2) (-sin(e^(x^5 sinx)))e^(x^5 sinx)* d/(dx) (x^5sinx)`

Use the product rule, `u = x^5, u' = 5x^4, v = sinx, v' = cos x`:

`y' = 1/2(cos(e^(x^5sinx)))^(-1/2) (-sin(e^(x^5 sinx)))e^(x^5 sinx) (x^5 cos x + 5 x^4 sin x)`

Rearrange and factor (GCF = `x^4`):

`y' = (-x^4e^(x^5sinx)sin(e^(x^5sinx))(xcosx + 5sinx))/(2sqrt(cos(e^(x^5sinx)))`