We are given the function $f(x) = -2 - x^2$ and asked to express the value of

$\frac{f(x+h) - f(x)}{h}$

in its simplest form. This expression is a difference quotient, often used to calculate the derivative of a function.

Let's solve this step by step.

1. Find $f(x+h)$: Start by substituting $x+h$ into the function $f(x) = -2 - x^2$.

   $f(x+h) = -2 - (x+h)^2$ Expand $(x+h)^2$: $(x+h)^2 = x^2 + 2xh + h^2$ So, $f(x+h) = -2 - (x^2 + 2xh + h^2) = -2 - x^2 - 2xh - h^2$

2. Subtract $f(x)$: Now, subtract $f(x) = -2 - x^2$ from $f(x+h)$: $f(x+h) - f(x) = \left( -2 - x^2 - 2xh - h^2 \right) - \left( -2 - x^2 \right)$ Simplifying: $f(x+h) - f(x) = -2 - x^2 - 2xh - h^2 + 2 + x^2$ The $-2$ and $+2$ cancel out, and $-x^2$ and $+x^2$ cancel out, leaving: $f(x+h) - f(x) = -2xh - h^2$

3. Divide by $h$: Finally, divide the result by $h$: $\frac{f(x+h) - f(x)}{h} = \frac{-2xh - h^2}{h}$ Simplify by dividing each term by $h$: $\frac{f(x+h) - f(x)}{h} = -2x - h$

Thus, the simplified form of the difference quotient is:

$\frac{f(x+h) - f(x)}{h} = -2x - h$

Would you like further explanation or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the purpose of the difference quotient in calculus?
2. How would you calculate the derivative of the function $f(x) = -2 - x^2$ directly?
3. What happens to the difference quotient as $h$ approaches 0?
4. Can you explain the geometric interpretation of the difference quotient?
5. How is the difference quotient related to the slope of a secant line?

Tip: The difference quotient is a critical step in deriving the concept of a derivative, which measures the instantaneous rate of change of a function.