Derivative of sqrt(x^2)

The solution

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   ____
  /  2 
\/  x

$$\sqrt{x^{2}}$$

  /   ____\
d |  /  2 |
--\\/  x  /
dx

$$\frac{d}{d x} \sqrt{x^{2}}$$

Transform

Detail solution

Let $u = x^{2}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{2}$ :
1. Apply the power rule: $x^{2}$ goes to $2 x$
The result of the chain rule is:

$\frac{x}{\left|{x}\right|}$

The answer is:

$\frac{x}{\left|{x}\right|}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

|x|
---
 x

$$\frac{\left|{x}\right|}{x}$$

Simplify

The second derivative [src]

  |x|          
- --- + sign(x)
   x           
---------------
       x

$$\frac{\operatorname{sign}{\left(x \right)} - \frac{\left|{x}\right|}{x}}{x}$$

Simplify

The third derivative [src]

  /|x|   sign(x)                \
2*|--- - ------- + DiracDelta(x)|
  |  2      x                   |
  \ x                           /
---------------------------------
                x

$$\frac{2 \left(\delta\left(x\right) - \frac{\operatorname{sign}{\left(x \right)}}{x} + \frac{\left|{x}\right|}{x^{2}}\right)}{x}$$

Simplify

The graph

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Identical expressions

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Derivative of sqrt(x^2)

The graph:

Piecewise:

The solution