Question

How do you use the chain rule to differentiate `y=sin4x^3`?

Answer 1

`12x^2 cos 4x^3`

Explanation:

`y=sin4x^3`

The derivative of `y`, denoted by `y'` will be `(dy)/(dx)=(dy)/(du). (du)/(dx)` (this is the Chain Rule)

Letting `u=4x^3=>y=sin u` and `(dy)/(du)=cos u`

`u=4x^3=>(du)/(dx)=12x^2`

So, your derivative, `y'=12x^2 cos u`

Substituting back the `u=>y'=12x^2 cos 4x^3`

I personally use formulas for these kind of problems (if of course they don't specify) because `1.` It's quicker `2.` It helps you check your work `3.` Why not? We have to memorize some basic formulas for the future anyways (^_^)

Using the formula `color (red) ((sin u)'=u' cos u)`

`=>y'=(sin u)'=12x^2 cos 4x^3`

Hope this helps :)