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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.

Given the function f(x) = -2 - x^2, express the value of (f(x+h) - f(x)) / h in simplest form.

Learn how to simplify the difference quotient for the function f(x) = -2 - x^2. This calculus problem involves expanding quadratic terms and simplifying the expression step-by-step. Suitable for high school students in Grades 10-12.

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