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:

`(d)/(dx)\lbrack x^3\rbrack+(d)/(dx)\lbrack2x^2\rbrack+(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`