Use the alternative form of the derivative to find the derivative at

*x* = *c*,

if it exists. (If an answer does not exist, enter DNE.)

*f*(*x*) = *x*2 − 6,

*c* = 7


To find the derivative of the function \( f(x) = x^2 - 6 \) at \( x = c \), where \( c = 7 \), we will use the alternative form of the derivative, which is defined as:

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

### Step 1: Identify the function and the point

Given:
- \( f(x) = x^2 - 6 \)
- \( c = 7 \)

First, calculate \( f(c) \):

\[
f(7) = 7^2 - 6 = 49 - 6 = 43
\]

### Step 2: Set up the limit definition

Now, apply the alternative form of the derivative:

\[
f'(7) = \lim_{x \to 7} \frac{f(x) - f(7)}{x - 7}
\]

Substitute the expressions for \( f(x) \) and \( f(7) \):

\[
f'(7) = \lim_{x \to 7} \frac{(x^2 - 6) - 43}{x - 7}
\]

Simplify the expression inside the limit:

\[
f'(7) = \lim_{x \to 7} \frac{x^2 - 49}{x - 7}
\]

### Step 3: Factor the numerator

Notice that the numerator \( x^2 - 49 \) is a difference of squares:

\[
x^2 - 49 = (x - 7)(x + 7)
\]

Substitute this back into the limit:

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

### Step 4: Simplify the expression

Cancel out the common factor \( x - 7 \) (assuming \( x \neq 7 \)):

\[
f'(7) = \lim_{x \to 7} (x + 7)
\]

### Step 5: Evaluate the limit

Now, substitute \( x = 7 \):

\[
f'(7) = 7 + 7 = 14
\]

### Conclusion

The derivative of \( f(x) = x^2 - 6 \) at \( x = 7 \) is:

\[
\boxed{14}
\]

This derivative exists and equals 14.

---

Would you like more details, or do you have any questions?

### Additional Questions:
1. What is the general derivative of the function \( f(x) = x^2 - 6 \)?
2. How would you use the limit definition of the derivative to find the derivative at any point \( x = c \)?
3. Can you use the power rule to confirm the result obtained using the alternative form?
4. How does the derivative at a point relate to the tangent line of the function at that point?
5. What happens to the derivative if \( f(x) \) is a constant function, like \( f(x) = -6 \)?

#### Tip:
When using the limit definition to find derivatives, always look for opportunities to simplify the expression before substituting values.

Use the alternative form of the derivative to find the derivative at x = c, if it exists. (If an answer does not exist, enter DNE.) f(x) = x^2 − 6, c = 7

Discover how to find the derivative of the quadratic function f(x) = x^2 - 6 at x = 7 using the limit definition. This step-by-step guide explains key calculus concepts such as limits and derivatives, providing a clear solution. Ideal for high school students in Grades 11-12.

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