Chapter 14.4, Problem 23AYU

To determine

To find: The derivative of $f\left(x\right) = x^2 - 3$ at 0.

Answer to Problem 23AYU

0

Explanation of Solution

Given:

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

$c = 0$

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, $f\left(x\right) = x^2 - 3$ and $c = 0$.

$f\left(c\right) = f\left(0\right) = 0^2 - 3 = 0 - 3 = -3$

Therefore,

$f^{\prime }\left(0\right) = \underset{x\to 0}{\lim }\frac{f\left(x\right) - f\left(0\right)}{x - 0}$

$f^{\prime }\left(0\right) = \underset{x\to 0}{\lim }\left[\frac{x^2 - 3 - \left(-3\right)}{x - 0}\right]$

$f^{\prime }\left(0\right) = \underset{x\to 0}{\lim }\left(\frac{x^2 - 3 + 3}{x}\right)$

$f^{\prime }\left(0\right) = \underset{x\to 0}{\lim }\left(\frac{x^2}{x}\right)$

$f^{\prime }\left(0\right) = \underset{x\to 0}{\lim }\left(x\right)$

$f^{\prime }\left(0\right) = 0$