Question

Use the alternative form of the derivative to find the derivative at x = c (if it exists). (If the derivative does not exist at c, enter UNDEFINED.)

f(x) = x3 + 4x2 + 5, c = −4
f '(−4) =

Answer 1

To find the derivative of the function f(x) = x3 + 4x2 + 5 at c = -4, apply the alternative form of the derivative by using the definition of the derivative at a point, evaluating f(-4 + h) and f(-4), and taking the limit of the difference quotient as h approaches zero.

Step-by-step explanation:

To find the derivative of f(x) = x3 + 4x2 + 5 at x = c, specifically at c = -4, using the alternative form of the derivative, we follow these steps:

1. First, recall the definition of the derivative at a point, f'(c) = lim (h -> 0) [f(c+h) - f(c)] / h. The alternative form involves calculating this limit.
2. Apply this definition to the given function f(x) at c = -4. This requires evaluating f(-4 + h) and f(-4), then finding the limit as h approaches zero.
3. Substitute c = -4 into the function to get f(-4) = (-4)3 + 4(-4)2 + 5, which simplifies to 5.
4. Expand f(-4 + h) to get [(-4 + h)3 + 4(-4 + h)2 + 5].
5. Take the limit of the difference quotient as h approaches zero: lim (h -> 0) { [(-4 + h)3 + 4(-4 + h)2 + 5] - 5 } / h.
6. Simplify the expression and resolve the limit to obtain the derivative f'(-4).

If any step results in an indeterminate form or does not resolve to a real number, then the derivative at c = -4 does not exist, and the answer is UNDEFINED.