Question

Find the derivative y'=dy/dx of the following function using the derivative rules?

Answer 1

`-24cos^3(6x)sin(6x)`

Explanation:

...which you get by applying the chain rule -you have to apply it more than once.

If `f(x) = u(v(x))`, then `f'(x) = (du)/(dv) * (dv)/(dx)`

So, I solved this one from the 'inside out'.

First, I found `d/dx(cos(6x))`.

Here `u(v) = cos(v)`, and `v(x) = 6x`

So `(dv)/(dx)` is 6, and `(du)/(dv) = -sin(v)`

By the chain rule, then, `d/dx(cos(6x)) = -6sin(6x)`

Now jump one level out: We have `u(v) = v^4`, and v(x) = cos(6x)#

Once again using the chain rule: `(du)/(dv) = 4v^3` and `(dv)/(dx) = -6sin(6x)`, which we solved for just now.

So `d/(dx) (cos^4(6x)) = (4)(-6)cos^3(6x)sin(6x) = -24cos^3(6x)sin(6x)`

GOOD LUCK