Question

What´s is the derivative of y=(x)(sinx)^3 ?

Answer 1

`sin^2(x)*(sin(x) + 3x*cos(x))`

Explanation:

Firstly, this is a function that is multiplied by another function.
So we'd initially use the product rule.

`d/dx [x*sin(x)] = d/dx[x]*sin^3(x) + x*d/dx[sin^3(x)]`

The derivative of x is 1:

`1*sin^3(x) + x*d/dx[sin^3(x)]`

Then you need to realize that `sin^3(x)` is just an application of the chain rule with the power rule which states that `d/dx[u(x)^n] = n * u(x)^(n-1) * u'(x)`

Apply:
`d/dx[sin^3(x)] = 3 * sin^2(x) * cos(x)`

Now we have the derivative of `sin^3(x)` so we can finally solve the question:

`sin^3(x) + x*d/dx[sin^3(x)] = sin^3(x) + x*3 * sin^2(x) * cos(x)`

Which simplifies to: (factoring out `sin^2(x)`)

`sin^2(x)*(sin(x) + 3x*cos(x))`