Question

Question #65ded

Answer 1

`dy/dx=color(blue)((sqrt(2)*cos(sqrt(x)))/(4sqrt(sin(sqrt(x)))*sqrt(x)))`

Explanation:

Derivative of `sqrt(2sin(sqrt(x))`

`sqrt(2sin(sqrt(x))) =(2sin(x^(1/2)))^(1/2)`

Using the chain rule:

`dy/dx=dy/(du) * (du)/(dw) *(dw)/dx`

Let `w=x^(1/2)`

Let `u=2sin(w)`

`dy/(du)((u)^(1/2))=1/2(u)^(-1/2)`

`(du)/(dw)(u)=2cos(w)`

`(dw)/dx(w)=1/2x^(-1/2)`

`dy/dx=1/2(u)^(-1/2) * 2cos(w) *1/2x^(-1/2)`

`dy/dx=1/2(2sin(x^(1/2)))^(-1/2) * 2cos(x^(1/2)) *1/2x^(-1/2)`

`dy/dx=1/(2(2sin(x^(1/2)))^(1/2)) * (2cos(x^(1/2)))/(2x^(1/2))`

`dy/dx=(2cos(x^(1/2)))/(2sqrt(2sin(sqrt(x)))(2sqrt(x)))`

`dy/dx=(2cos(x^(1/2)))/(4sqrt(2)sqrt(sin(sqrt(x)))sqrt(x))`

`dy/dx=color(blue)((sqrt(2)*cos(sqrt(x)))/(4sqrt(sin(sqrt(x)))*sqrt(x)))`