What are the steps behind differentiating and simplifying the following:

y = sin^3x

In getting y' = 3sin^2xcosx why do we not get cos^2x?

By using the chain rule.

\frac{d}{dx}[sin^{3}(x)]=3sin^{2}(x)cos(x)

The derivative of the inside times the derivative of the outside.

The derivative of the outside is 3sin^{2}(x)

The derivative of the inside is cos(x)

We get the cos(x) by just differentiating sin(x).

You can also have it with the product rule.

You know that \frac{d(sin^2(x))}{dx} = 2sin(x)cos(x) ?

Then put it like that: sin^3(x) = sin^2(x)sin(x)

Use the product rule here;

\frac{d(sin^2(x)sin(x))}{dx} = sin^2(x)cos(x) + sin(x).2sin(x)cos(x) = sin^2(x)cos(x) + 2sin^2(x)cos(x) = 3sin^2(x)cos(x)