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.

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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.