Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of e^x^x
Derivative of sqrt4
Derivative of cos(x)/e^x
Derivative of acos(1/x)
Identical expressions
ln(x+sqrt(four +x^ two))
ln(x plus square root of (4 plus x squared ))
ln(x plus square root of (four plus x to the power of two))
ln(x+√(4+x^2))
ln(x+sqrt(4+x2))
lnx+sqrt4+x2
ln(x+sqrt(4+x²))
ln(x+sqrt(4+x to the power of 2))
lnx+sqrt4+x^2
Similar expressions
ln(x+sqrt(4-x^2))
ln(x-sqrt(4+x^2))

The derivative of the function/ ln(x+sqrt(4+x^2))

Derivative of ln(x+sqrt(4+x^2))

The solution

You have entered [src]

   /       ________\
   |      /      2 |
log\x + \/  4 + x  /

$$\log{\left(x + \sqrt{x^{2} + 4} \right)}$$

log(x + sqrt(4 + x^2))

Detail solution

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

$\frac{\frac{x}{\sqrt{x^{2} + 4}} + 1}{x + \sqrt{x^{2} + 4}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

 /                 2              \ 
 |/         x     \            2  | 
 ||1 + -----------|           x   | 
 ||       ________|    -1 + ------| 
 ||      /      2 |              2| 
 |\    \/  4 + x  /         4 + x | 
-|------------------ + -----------| 
 |        ________        ________| 
 |       /      2        /      2 | 
 \ x + \/  4 + x       \/  4 + x  / 
------------------------------------
                 ________           
                /      2            
          x + \/  4 + x

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

Simplify

The third derivative [src]

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

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

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of ln(x+sqrt(4+x^2))

The graph:

Piecewise:

The solution