Let y=x^{cos(x)} and z=sin(y)=sin(x^{cos(x)}). You'll need to first find dy/dx before using the Chain Rule to find dz/dx=dz/dy*dy/dx.
Find dy/dx by using logarithmic differentiation (which also involves implicit differentiation). Since y=x^{cos(x)}, we can also write ln(y)=ln(x^{cos(x)})=cos(x)ln(x), where the last equality is a property of logarithms.
Now differentiate both sides of this last equation with respect to x (keeping in mind that y is a function of x) and using the Chain Rule and Product Rule to get 1/y*dy/dx=-sin(x)ln(x)+cos(x)/x so that dy/dx=y(cos(x)/x-sin(x)ln(x))=x^{cos(x)}(cos(x)/x-sin(x)ln(x))
Since dz/dy=cos(y)=cos(x^{cos(x)}), the Chain Rule (of the form in the first paragraph) can now be brought to bear to conclude that
dz/dx=cos(x^{cos(x)})*x^{cos(x)}(cos(x)/x-sin(x)ln(x)).
That's the answer.
Here's what the graph of sin(x^(cos(x))) looks like (for 0 < x < 6\pi), in case you were wondering:
![enter image source here]()
