Let's break down and solve this problem step-by-step.

The function $f$ is defined as: $f(x) = -2x^2$

You are asked to calculate and simplify the following expression: $\frac{f(-4 + h) - f(-4)}{h}$

Step 1: Calculate $f(-4 + h)$

$f(-4 + h) = -2(-4 + h)^2$ Expanding $(-4 + h)^2$: $(-4 + h)^2 = (-4)^2 + 2(-4)(h) + h^2 = 16 - 8h + h^2$ So: $f(-4 + h) = -2(16 - 8h + h^2) = -32 + 16h - 2h^2$

Step 2: Calculate $f(-4)$

$f(-4) = -2(-4)^2 = -2(16) = -32$

Step 3: Simplify the expression $\frac{f(-4 + h) - f(-4)}{h}$

Substitute the values from steps 1 and 2: $\frac{(-32 + 16h - 2h^2) - (-32)}{h} = \frac{-32 + 16h - 2h^2 + 32}{h} = \frac{16h - 2h^2}{h}$ Simplify: $\frac{16h - 2h^2}{h} = 16 - 2h$

Step 4: Calculate the limit as $h \to 0$

$\lim_{h \to 0} (16 - 2h) = 16$

Thus, the derivative $f'(-4)$ is: $f'(-4) = 16$

Would you like more details on any step or further clarification?

Related Questions:

1. How is the derivative defined in terms of limits?
2. What happens if we change the value at which we calculate the derivative?
3. Can you generalize the derivative for any $f(x) = -2x^2$?
4. How does the limit process help in calculating derivatives?
5. How does this method relate to the power rule in differentiation?

Tip:

Always expand and simplify expressions carefully when applying the limit definition of a derivative.