Derivative of sqrt(x*x+4)

You have entered [src]

  _________
\/ 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)

Plot the graph f'(x)

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sqrt(x*x+4)

The graph:

Piecewise:

The solution