Derivative of sqrt(x*x-4)

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  _________
\/ x*x - 4

$$\sqrt{x x - 4}$$

sqrt(x*x - 4)

Detail solution

Let $u = x x - 4$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x x - 4\right)$ :
1. Differentiate $x x - 4$ term by term:
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    $g{\left(x \right)} = x$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    The result is: $2 x$
  2. The derivative of the constant $-4$ is zero.
  The result is: $2 x$
The result of the chain rule is:

$\frac{x}{\sqrt{x x - 4}}$
Now simplify:

$\frac{x}{\sqrt{x^{2} - 4}}$

The answer is:

$\frac{x}{\sqrt{x^{2} - 4}}$

The graph

Plot the graph f(x)

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The first derivative [src]

     x     
-----------
  _________
\/ x*x - 4

$$\frac{x}{\sqrt{x x - 4}}$$

Simplify

The second derivative [src]

        2   
       x    
1 - ------- 
          2 
    -4 + x  
------------
   _________
  /       2 
\/  -4 + x

$$\frac{- \frac{x^{2}}{x^{2} - 4} + 1}{\sqrt{x^{2} - 4}}$$

Simplify

The third derivative [src]

    /         2  \
    |        x   |
3*x*|-1 + -------|
    |           2|
    \     -4 + x /
------------------
            3/2   
   /      2\      
   \-4 + x /

$$\frac{3 x \left(\frac{x^{2}}{x^{2} - 4} - 1\right)}{\left(x^{2} - 4\right)^{\frac{3}{2}}}$$

Simplify

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Derivative of sqrt(x*x-4)

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The solution