Chapter 14.4, Problem 26AYU

To determine

To find: The derivative of $f\left(x\right) = 3x^2 - 4x$ at 2.

Answer to Problem 26AYU

8

Explanation of Solution

Given:

$f\left(x\right) = 3x^2 - 4x$

$c = 2$

Formula used:

The derivative of $y = f\left(x\right)$ at $c$ is given by:

$f^{\prime }\left(c\right) = \underset{x\to c}{\lim }\frac{f\left(x\right) - f\left(c\right)}{x - c}$

Calculation:

Here, $f\left(x\right) = 3x^2 - 4x$ and $c = 2$.

$f\left(c\right) = f\left(2\right) = 3{\left(2\right)}^2 - 4\left(2\right) = 12 - 8 = 4$

Therefore,

$f^{\prime }\left(2\right) = \underset{x\to 2}{\lim }\frac{f\left(x\right) - f\left(2\right)}{x - 2}$

$f^{\prime }\left(2\right) = \underset{x\to 2}{\lim }\left(\frac{3x^2 - 4x - 4}{x - 2}\right)$

$f^{\prime }\left(2\right) = \underset{x\to 2}{\lim }\frac{\left(3x + 2\right)\left(x - 2\right)}{\left(x - 2\right)}$

$f^{\prime }\left(2\right) = \underset{x\to 2}{\lim }\left(3x + 2\right)$

$f^{\prime }\left(2\right) = 3\left(2\right) + 2$

$f^{\prime }\left(2\right) = 6 + 2$

$f^{\prime }\left(2\right) = 8$