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

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Answer 1 · 2016-09-01T19:30:40

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

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

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