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Derivative of:
Derivative of y=ln(x+sqrt(x^2+4))
Derivative of ((1+3z)/3*z)(3-z)
Derivative of (3x+4)²
Derivative of (4x^3-3x^2)/(2x-1)^2
Identical expressions
y=ln(x+sqrt(x^ two + four))
y equally ln(x plus square root of (x squared plus 4))
y equally ln(x plus square root of (x to the power of two plus four))
y=ln(x+√(x^2+4))
y=ln(x+sqrt(x2+4))
y=lnx+sqrtx2+4
y=ln(x+sqrt(x²+4))
y=ln(x+sqrt(x to the power of 2+4))
y=lnx+sqrtx^2+4
Similar expressions
y=ln(x+sqrt(x^2-4))
y=ln(x-sqrt(x^2+4))

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

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

The solution

You have entered [src]

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

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

  /   /       ________\\
d |   |      /  2     ||
--\log\x + \/  x  + 4 //
dx

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

Transform

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. Apply the power rule: $x^{2}$ goes to $2 x$
      2. The derivative of the constant $4$ is zero.
      The result is: $2 x$
    The result of the chain rule is:
    
    $\frac{x}{\sqrt{x^{2} + 4}}$
  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     
    \/  x  + 4 
---------------
       ________
      /  2     
x + \/  x  + 4

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

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

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

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

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

Simplify

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

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The solution