Find the derivative of the function f(x) =x^2 -3 using the limit process

To find the derivative of the function \( f(x) = x^2 - 3 \) using the limit definition of a derivative, we will apply the formula:

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

### Step-by-step process:

1. **Substitute \( f(x) = x^2 - 3 \) into the limit definition:**

\[
f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - 3 - (x^2 - 3)}{h}
\]

2. **Expand \( (x+h)^2 \):**

\[
(x+h)^2 = x^2 + 2xh + h^2
\]

Now substitute this into the equation:

\[
f'(x) = \lim_{h \to 0} \frac{(x^2 + 2xh + h^2 - 3) - (x^2 - 3)}{h}
\]

3. **Simplify the expression:**

First, cancel out common terms:

\[
f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 3 - x^2 + 3}{h}
\]

This simplifies to:

\[
f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h}
\]

4. **Factor out \( h \) from the numerator:**

\[
f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h}
\]

Cancel the \( h \):

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

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

\[
f'(x) = 2x
\]

Thus, the derivative of \( f(x) = x^2 - 3 \) is:

\[
f'(x) = 2x
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you find the derivative of \( f(x) = x^3 \) using the limit process?
2. What is the derivative of \( f(x) = \frac{1}{x} \) using the limit definition?
3. Can you explain the difference between the power rule and the limit process for finding derivatives?
4. What does the derivative of a constant function look like?
5. How does the derivative of a quadratic function relate to its graph?

**Tip:** The limit definition of a derivative provides an intuitive understanding of how instantaneous rates of change are calculated.

Find the derivative of the function f(x) = x^2 - 3 using the limit process

Learn how to find the derivative of the function f(x) = x^2 - 3 using the limit process. This step-by-step solution covers the key concepts of calculus, including the limit definition of derivatives. Suitable for students in Grades 11-12 studying calculus.

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