9–61. Evaluate and simplify y'.


y = e^sin (cosx)

Verified step by step guidance

1. Step 1: Identify the function y = e^{\sin(\cos x)} and recognize that it is a composition of functions, which will require the use of the chain rule to differentiate.
2. Step 2: Apply the chain rule. The chain rule states that if you have a composite function y = f(g(x)), then the derivative y' = f'(g(x)) * g'(x).
3. Step 3: Differentiate the outer function e^{u} with respect to u, where u = \sin(\cos x). The derivative of e^{u} with respect to u is e^{u}.
4. Step 4: Differentiate the inner function \sin(\cos x) with respect to x. This requires using the chain rule again: first differentiate \sin(v) with respect to v, where v = \cos x, and then differentiate \cos x with respect to x.
5. Step 5: Combine the derivatives from Steps 3 and 4. Multiply the derivative of the outer function by the derivative of the inner function to find y'.