https://file.aiquickdraw.com/mathbot/file/17246683463712ktopqk5.jpg

The formula provided in the image is the definition of the derivative using limits:

\[
f'(x) = \lim_{z \to x} \frac{f(z) - f(x)}{z - x}
\]

Let's calculate the derivative of the given function \( F(a) = a^3 + 2a - 7 \) using this limit definition.

### Step 1: Define \( f(z) \) and \( f(x) \)
Given \( F(a) = a^3 + 2a - 7 \), we'll substitute \( z \) and \( x \) into this function:

\[
f(z) = z^3 + 2z - 7
\]
\[
f(x) = x^3 + 2x - 7
\]

### Step 2: Apply the Limit Definition
Substitute \( f(z) \) and \( f(x) \) into the limit definition of the derivative:

\[
f'(x) = \lim_{z \to x} \frac{(z^3 + 2z - 7) - (x^3 + 2x - 7)}{z - x}
\]

### Step 3: Simplify the Expression
First, simplify the expression in the numerator:

\[
f'(x) = \lim_{z \to x} \frac{z^3 + 2z - 7 - x^3 - 2x + 7}{z - x}
\]

Combine like terms:

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

Notice that \( z^3 - x^3 \) can be factored using the difference of cubes, and \( 2z - 2x \) can be factored as well:

\[
f'(x) = \lim_{z \to x} \frac{(z - x)(z^2 + zx + x^2) + 2(z - x)}{z - x}
\]

Factor out \( z - x \) from the numerator:

\[
f'(x) = \lim_{z \to x} \frac{(z - x)(z^2 + zx + x^2 + 2)}{z - x}
\]

Cancel out \( z - x \) from the numerator and denominator:

\[
f'(x) = \lim_{z \to x} (z^2 + zx + x^2 + 2)
\]

### Step 4: Evaluate the Limit
Finally, substitute \( z = x \) into the simplified expression:

\[
f'(x) = x^2 + x^2 + x^2 + 2 = 3x^2 + 2
\]

### Conclusion
This result confirms the earlier derivative we found using standard differentiation:

\[
F'(a) = 3a^2 + 2
\]

Would you like more details or have any questions?

Here are some related questions to explore:
1. How does the limit definition of a derivative relate to instantaneous rate of change?
2. Can the derivative of a function be found using the difference quotient?
3. How would you differentiate a function involving both polynomials and trigonometric terms using this limit process?
4. What are some common mistakes to avoid when applying the limit definition of a derivative?
5. How does this method extend to higher dimensions, such as partial derivatives?

**Tip:** When dealing with derivatives, practicing the limit definition helps in understanding the core concept of how a derivative measures change.

Learn how to calculate the derivative of a polynomial function using the limit definition step-by-step. This example problem demonstrates the application of the limit definition of derivatives to find the derivative of a polynomial function. Suitable for undergraduate students studying calculus.

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