Question

Use the alternative form of the derivative to find the derivative at x = c (if it exists). (If the derivative does not exist at c, enter UNDEFINED.)

Answer 1

Given:

$f(x)=x^3+2x^2+8,c=-2$

Let's use the alternate form of the derivative to find the derivative at x = c.

Apply the sum rule:

$\frac{d}{dx}\lbrack x^3\rbrack+\frac{d}{dx}\lbrack2x^2\rbrack+\frac{d}{dx}\lbrack8\rbrack$

Apply power rule to differentiate:

$\begin{gathered} 3x^2+4x+0 \\ \\ f^{\prime}(x)=3x^2+4x \end{gathered}$

Now, let's solve f'(-2).

To solve for f'(-2), substitute -2 for x in the derivative and evaluate:

$\begin{gathered} f^{\prime}(-2)=3(-2)^2+4(-2) \\ \\ f^{\prime}(-2)=3(4)+(-8) \\ \\ f^{\prime}(-2)=12-8 \\ \\ f^{\prime}(-2)=4 \end{gathered}$

ANSWER:

$f^{\prime}(-2)=4$