How do you differentiate sin sin x?

To differentiate y=sin(sin(x)) you need to use the chain rule.

A common way to remember the chain rule is "derivative of the outside, keep the inside, derivative of the inside".

First, you take the derivative of the outside. The derivative of sin is cos.

Then, you keep the inside, so you keep sin(x).

Then, you multiple by the derivative of the inside. Again, the derivative of sinx is cosx.

In the end, you get y'=cos(sin(x))cos(x))