Question

What is the derivative of `f(x)=cos^6(2x+5)`?

Answer 1

`dy/dx=-12sin(2x+5)cos(2x+5)`

There are trig identities you could play with to simplify this solution further.

Explanation:

Using 'old fashioned' notation

Let `u= 2x+5 => (du)/dx = 2`

Let `v=cos(u) =>(dv)/(du)=-sin(u)`

Let `y=v^6 => (dy)/(dv)=6v^5`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Consider the product:

`dy/(cancel(dv))xx(cancel(dv))/(cancel(du))xx(cancel(du))/(dx) = dy/dx`

Putting this all together:

`dy/dx=6v^5xx(-sin(u))xx2`

So by substitution we have:

`dy/dx=6cos^5(2x+5)xx (-sin(2x+5))xx2`

`dy/dx=-12sin(2x+5)cos(2x+5)`
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