Question

How do you differentiate `f(x)=(1+cos^2x)^6`?

Answer 1

`f'(x)=-6sin2x(1+cos^2x)^5`

Explanation:

differentiate using the `color(blue)"chain rule"`

`"Given " f(x)=g(h(x))" then"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(f'(x)=g'(h(x))xxh'(x))color(white)(2/2)|)))`

`rArrf'(x)=6(1+cos^2x)^5xxd/dx(1+cos^2x)to(1)`

`"Using the " color(blue)"chain rule " "on " (1+cos^2x)`

`d/dx(1+cos^2x)=2cosxxd/dx(cosx)`

`color(white)(d/dx(1+cos^2x))=-2cosxsinxlarr( -sin2x)`

`"Returning to " (1)`

`f'(x)=-6sin2x(1+cos^2x)^5`