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2.Absoulte Value



1. Logarithmic
@2. Absoulte Value
3. The Square Root

The absolute value of $x$ is denoted as $|x|$. We have that $|x|$ is the largest of the two values $x$ and $-x$. Or we can write is as $$ |x| = \begin{cases} x & x \geq 0 \\ -x & x \lt 0 \end{cases} $$ We have that $|x| \geq 0$. We can plot it with the following python3 code

import numpy as np import matplotlib.pyplot as plt def f(x): return np.abs(x) x = [float(x) / 10.0 for x in range(-1 * 100,100)] y = [f(x) for x in x] plt.plot(x,y)

For which we get the plot as seen in Figure 1.

Figure 1: plot of the absolute value $|x|$.

The derivative of the absolute value

The derivative is given as $$ |x|' = \frac{x}{|x|} $$ This looks more scary than it is. It just states that if $x \lt 0$, then $|x|' = -1$. If $x \gt 0$, then $|x|' = 1$. Note that the derivative is not defined when $x = 0$. We can derive the derivative. First rewrite $$ |x| = \sqrt{x^2} $$ We use the chain rule to obtain $$ \sqrt{x^2}' = \frac{1}{2 \sqrt{x^2}} \cdot 2x = \frac{x}{\sqrt{x^2}} $$ We substitute in order to obtain $$ |x|' = \frac{x}{|x|} $$

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