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'.

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this context, we need to apply differentiation rules to the function y = e^sin(cos(x)) to find y'. This involves using the chain rule and product rule, as the function is a composition of multiple functions.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. In this case, we will apply the chain rule to differentiate e^sin(cos(x)).

Exponential Functions

Exponential functions are functions of the form y = a^x, where a is a constant and x is the variable. The derivative of an exponential function, particularly when the base is e, is unique because it equals the function itself multiplied by the derivative of the exponent. Understanding how to differentiate e^u, where u is a function of x, is crucial for solving the given problem.