How do you differentiate y=sin(cos(x))?

To solve this question we will use the chain rule, as we can see that we have one function being applied to another, i.e sin is being applied to cos(x).

This means we are able to replace the original function (cos(x)) with a dummy variable, in this case we will use u.

i.e y=sin(u), u=cos(x).

Firstly, we will differentiate y with respect to u:

y=sin(u).

sin differentiates to cos, so therefore we have:

dy/du=cos(u).

Secondly, we will differentiate u with respect to x:

u=cos(x).

cos when differentiated becomes -sin, so therefore we have:

du/dx=-sin(x).

We will now use the chain rule:

i.e dy/dx=dy/du*du/dx.

Replacing the differentials we found earlier, means that:

dy/dx=cos(u)*-sin(x).

We will now replace u with cos(x).

This gives:

dy/dx = cos(cos(x))*-sin(x).