Final answer:
To find f'(8), we use the chain rule and given values of g(8) and g'(8) to calculate the derivative of f(x). The derivative of f(x) is -4, so f'(8) = -4.
Step-by-step explanation:
To find the value of f'(8), we need to use the chain rule. The chain rule states that if we have a composite function f(g(x)), then the derivative of f(g(x)) with respect to x is f'(g(x)) * g'(x). In this case, f(x) = e^(-g(x)), so we need to find the derivative of e^(-g(x)).
Let's start by finding the derivative of the inner function g(x). We are given that g(8) = 0 and g'(8) = -4. So g(x) = 0 and g'(x) = -4 everywhere.
Now, let's find the derivative of f(x). The derivative of e^x is e^x. So f'(x) = e^(-g(x)) * g'(x) = e^0 * (-4) = -4.
Therefore, f'(8) = -4.