let f(x)=x^2-2. finish the steps that find the instantaneous rate of change of f at x=1 using limit definition.

To find the instantaneous rate of change of \( f(x) = x^2 - 2 \) at \( x = 1 \) using the limit definition, we'll use the formula for the derivative at a point:

\[
f'(1) = \lim_{h \to 0} \frac{f(1 + h) - f(1)}{h}
\]

### Step-by-Step Solution:

1. **Find \( f(1 + h) \):**

   Since \( f(x) = x^2 - 2 \), we substitute \( 1 + h \) into the function:

   \[
   f(1 + h) = (1 + h)^2 - 2
   \]

   Expand \( (1 + h)^2 \):

   \[
   (1 + h)^2 = 1^2 + 2(1)(h) + h^2 = 1 + 2h + h^2
   \]

   So,

   \[
   f(1 + h) = 1 + 2h + h^2 - 2 = h^2 + 2h - 1
   \]

2. **Find \( f(1) \):**

   Using the function \( f(x) = x^2 - 2 \), substitute \( x = 1 \):

   \[
   f(1) = 1^2 - 2 = 1 - 2 = -1
   \]

3. **Set up the difference quotient:**

   Substitute \( f(1 + h) \) and \( f(1) \) into the difference quotient formula:

   \[
   \frac{f(1 + h) - f(1)}{h} = \frac{(h^2 + 2h - 1) - (-1)}{h} = \frac{h^2 + 2h - 1 + 1}{h} = \frac{h^2 + 2h}{h}
   \]

4. **Simplify the expression:**

   Factor out \( h \) from the numerator:

   \[
   \frac{h(h + 2)}{h}
   \]

   Cancel the \( h \) in the numerator and denominator (assuming \( h \neq 0 \)):

   \[
   h + 2
   \]

5. **Take the limit as \( h \to 0 \):**

   Now, find the limit as \( h \to 0 \):

   \[
   \lim_{h \to 0} (h + 2) = 2
   \]

### Conclusion:

The instantaneous rate of change of \( f(x) = x^2 - 2 \) at \( x = 1 \) is \( 2 \).

---

Do you want more details, or have any other questions? Here are 5 related questions:

1. What does the limit definition of a derivative represent geometrically?
2. Can you use the same process to find the derivative at \( x = 2 \)?
3. What is the general formula for the derivative of \( f(x) = x^2 - 2 \)?
4. How would you find the instantaneous rate of change for a different function, like \( f(x) = x^3 - 2x \)?
5. Can you explain why the \( h \) term can be canceled in the simplification step?

**Tip:** The derivative of a function at a point gives the slope of the tangent line to the curve at that point.

Let f(x) = x^2 - 2. Finish the steps that find the instantaneous rate of change of f at x = 1 using limit definition.

Explore how to find the instantaneous rate of change of the function f(x) = x^2 - 2 at x = 1 using the limit definition of a derivative. This detailed step-by-step solution demonstrates key calculus concepts including limits, derivatives, and rate of change, suitable for high school or college calculus students.

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