Question

How do you find the derivative of `y=(1+cos^2x)^6` using the chain rule?

Answer 1

`dy/dx = -6sin2x (1 + cos^2 x)^5`

Explanation:

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

`d/dx [ f(g(x)) ] = f'(g(x)) . g'(x)`

here : f(g(x)) = `(1 + cos^2 x)^6`

`rArr f'(g(x)) = 6(1 + cos^2 x)^5`
`"-----------------------------------------------------"`

and g(x) =`(1 + cos^2 x) → g'(x) = 2cosx .d/dx(cosx)`

= 2cosx.(-sinx) = - 2sinxcosx = - sin2x
`"------------------------------------------------------"`

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

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