Question

How do you differentiate `f(x) = x/sqrt(sin^2(1+x^2)` using the chain rule?

Answer 1

`f'(x)=(sin(1+x^2)-2x^2cos(1+x^2))/sin^2(1+x^2)`

Explanation:

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

Use quotient rule and chain rule

`f=x, g=((sin(1+x^2))^2)^(1/2) = sin(1+x^2)`

`f'=1, g'=cos(1+x^2)*2x`

`f'(x)=(gf'-fg')/g^2`

`f'(x)=(sin(1+x^2)-2x^2cos(1+x^2))/sin^2(1+x^2)`