Question

use the alternative form of the derivative to find the derivative at x = c, if it exists. (if an a
f(x)=x² - 3, c = 6
f(6)=lim(x→c) (f(x)-f(c))/(x - c)
=lim(x→6) ((x² - 3)-( )×)/(x-(6)√)
=×

Explanation:

Step1: Compute f(6)

$f(6) = 6^2 - 3 = 33$

Step2: Substitute into numerator

$(x^2 - 3) - f(6) = (x^2 - 3) - 33 = x^2 - 36$

Step3: Factor numerator

$x^2 - 36 = (x - 6)(x + 6)$

Step4: Simplify the limit

$\lim_{x \to 6} \frac{(x - 6)(x + 6)}{x - 6} = \lim_{x \to 6} (x + 6)$

Step5: Evaluate the limit

$\lim_{x \to 6} (x + 6) = 6 + 6 = 12$

Answer:

12