Final answer:
The derivative f'(c) for the function f(x) = -3/(x + 5)^2 when c = 4 is computed using the chain rule. The result after differentiation and evaluation at c is f'(4) = 2/27.
Step-by-step explanation:
To find f′(c) when given that f(x) = −3/(x + 5)² and c = 4, we need to compute the derivative of the function f(x) and then evaluate it at x = 4. The first step involves finding the derivative f′(x) using the chain rule, because the function can be seen as an outer function −3/u² where u = (x + 5), which is itself a function of x.
The chain rule states that the derivative of a composed function like this one is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. So, first take the derivative of the outer function with respect to u, which is d/du(−3/u²) = 6/u³. Then, the derivative of the inner function u with respect to x is simply du/dx = 1. Applying the chain rule, we find:
f′(x) = (6/u³) × (du/dx) = 6/(x + 5)³
Now, to find f′(c) where c = 4, we substitute x with 4:
f′(4) = 6/(4 + 5)³ = 6/81 = 2/27
Thus, f′(4) = 2/27.