Derivative of √ln(x+√x+√x²+sin2x)

The solution

You have entered [src]

    ____________________________________
   /    /                 2           \ 
  /     |      ___     ___            | 
\/   log\x + \/ x  + \/ x   + sin(2*x)/

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

sqrt(log(x + sqrt(x) + (sqrt(x))^2 + sin(2*x)))

Detail solution

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

$\frac{2 \cos{\left(2 x \right)} + 2 + \frac{1}{2 \sqrt{x}}}{2 \left(\left(\left(\sqrt{x}\right)^{2} + \left(\sqrt{x} + x\right)\right) + \sin{\left(2 x \right)}\right) \sqrt{\log{\left(\left(\left(\sqrt{x}\right)^{2} + \left(\sqrt{x} + x\right)\right) + \sin{\left(2 x \right)} \right)}}}$
Now simplify:

$\frac{8 \sqrt{x} \cos^{2}{\left(x \right)} + 1}{4 \sqrt{x} \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right) \sqrt{\log{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)} \right)}}}$

The answer is:

$\frac{8 \sqrt{x} \cos^{2}{\left(x \right)} + 1}{4 \sqrt{x} \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right) \sqrt{\log{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)} \right)}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                              1                   x                       
                       1 + ------- + 2*cos(2*x) + -                       
                               ___                x                       
                           2*\/ x                                         
--------------------------------------------------------------------------
                                      ____________________________________
  /                 2           \    /    /                 2           \ 
  |      ___     ___            |   /     |      ___     ___            | 
2*\x + \/ x  + \/ x   + sin(2*x)/*\/   log\x + \/ x  + \/ x   + sin(2*x)/

$$\frac{2 \cos{\left(2 x \right)} + 1 + \frac{x}{x} + \frac{1}{2 \sqrt{x}}}{2 \left(\left(\left(\sqrt{x}\right)^{2} + \left(\sqrt{x} + x\right)\right) + \sin{\left(2 x \right)}\right) \sqrt{\log{\left(\left(\left(\sqrt{x}\right)^{2} + \left(\sqrt{x} + x\right)\right) + \sin{\left(2 x \right)} \right)}}}$$

Simplify

The second derivative [src]

 /                                              2                                          2               \ 
 |                      /      1               \                   /      1               \                | 
 |                      |4 + ----- + 4*cos(2*x)|                   |4 + ----- + 4*cos(2*x)|                | 
 |                      |      ___             |                   |      ___             |                | 
 |  1                   \    \/ x              /                   \    \/ x              /                | 
-|------ + 8*sin(2*x) + -------------------------- + ------------------------------------------------------| 
 |   3/2                  /  ___                 \     /  ___                 \    /  ___                 \| 
 \2*x                   2*\\/ x  + 2*x + sin(2*x)/   4*\\/ x  + 2*x + sin(2*x)/*log\\/ x  + 2*x + sin(2*x)// 
-------------------------------------------------------------------------------------------------------------
                                                       _____________________________                         
                           /  ___                 \   /    /  ___                 \                          
                         4*\\/ x  + 2*x + sin(2*x)/*\/  log\\/ x  + 2*x + sin(2*x)/

$$- \frac{8 \sin{\left(2 x \right)} + \frac{\left(4 \cos{\left(2 x \right)} + 4 + \frac{1}{\sqrt{x}}\right)^{2}}{2 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right)} + \frac{\left(4 \cos{\left(2 x \right)} + 4 + \frac{1}{\sqrt{x}}\right)^{2}}{4 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right) \log{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)} \right)}} + \frac{1}{2 x^{\frac{3}{2}}}}{4 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right) \sqrt{\log{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)} \right)}}}$$

Simplify

The third derivative [src]

                                                3                                                                                             3                                                         3                                                                        
                        /      1               \      / 1                \ /      1               \                   /      1               \                                  /      1               \                        / 1                \ /      1               \    
                        |4 + ----- + 4*cos(2*x)|    3*|---- + 16*sin(2*x)|*|4 + ----- + 4*cos(2*x)|                 3*|4 + ----- + 4*cos(2*x)|                                3*|4 + ----- + 4*cos(2*x)|                      3*|---- + 16*sin(2*x)|*|4 + ----- + 4*cos(2*x)|    
                        |      ___             |      | 3/2              | |      ___             |                   |      ___             |                                  |      ___             |                        | 3/2              | |      ___             |    
                 3      \    \/ x              /      \x                 / \    \/ x              /                   \    \/ x              /                                  \    \/ x              /                        \x                 / \    \/ x              /    
-32*cos(2*x) + ------ + ------------------------- + ----------------------------------------------- + ------------------------------------------------------- + -------------------------------------------------------- + ------------------------------------------------------
                  5/2                           2                /  ___                 \                                       2                                                         2                                  /  ___                 \    /  ___                 \
               2*x      /  ___                 \               2*\\/ x  + 2*x + sin(2*x)/               /  ___                 \     /  ___                 \     /  ___                 \     2/  ___                 \   4*\\/ x  + 2*x + sin(2*x)/*log\\/ x  + 2*x + sin(2*x)/
                        \\/ x  + 2*x + sin(2*x)/                                                      4*\\/ x  + 2*x + sin(2*x)/ *log\\/ x  + 2*x + sin(2*x)/   8*\\/ x  + 2*x + sin(2*x)/ *log \\/ x  + 2*x + sin(2*x)/                                                         
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                         _____________________________                                                                                                           
                                                                                                             /  ___                 \   /    /  ___                 \                                                                                                            
                                                                                                           8*\\/ x  + 2*x + sin(2*x)/*\/  log\\/ x  + 2*x + sin(2*x)/

$$\frac{\frac{3 \left(16 \sin{\left(2 x \right)} + \frac{1}{x^{\frac{3}{2}}}\right) \left(4 \cos{\left(2 x \right)} + 4 + \frac{1}{\sqrt{x}}\right)}{2 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right)} + \frac{3 \left(16 \sin{\left(2 x \right)} + \frac{1}{x^{\frac{3}{2}}}\right) \left(4 \cos{\left(2 x \right)} + 4 + \frac{1}{\sqrt{x}}\right)}{4 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right) \log{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)} \right)}} - 32 \cos{\left(2 x \right)} + \frac{\left(4 \cos{\left(2 x \right)} + 4 + \frac{1}{\sqrt{x}}\right)^{3}}{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right)^{2}} + \frac{3 \left(4 \cos{\left(2 x \right)} + 4 + \frac{1}{\sqrt{x}}\right)^{3}}{4 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right)^{2} \log{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)} \right)}} + \frac{3 \left(4 \cos{\left(2 x \right)} + 4 + \frac{1}{\sqrt{x}}\right)^{3}}{8 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right)^{2} \log{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)} \right)}^{2}} + \frac{3}{2 x^{\frac{5}{2}}}}{8 \left(\sqrt{x} + 2 x + \sin{\left(2 x \right)}\right) \sqrt{\log{\left(\sqrt{x} + 2 x + \sin{\left(2 x \right)} \right)}}}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of √ln(x+√x+√x²+sin2x)

The graph:

Piecewise:

The solution