e^(cos(x^3))
Well, we shall take it one by one. If you look closely, the expression is a function within a function within a function. We can look at the outermost term to be an exponential function. We know that deriv of e^x = e^x still. But x in here is cos (x^3), so the derivative of the outermost is e^(cos(x^3)) still.
Now the function inside the outermost function is cos (x^3). Since e is usually raised to just x, this function wherein x is also a function should also be differentiated. The derivative of cos (x^3) = -sin(x^3). Since we know that deriv of cos x = -sin x, where x in this case is x^3.
Next, we have the third layer of functions (x^3). This is easily differentiated. It's actually 3x^2. This would have to be the end of the layers of functions since the derivative of x is just 1. We have everything and the last step is to just multiply everything.
Therefore, the derivative of e^(cos(x^3)) using CR = e^(cos(x^3))(-sin(x^3))(3x^2).