Derivative of y=xln(sinsqrtxlnx+tg4x)

The solution

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x*log\sin\\/ x /*log(x) + tan(4*x)/

$$x \log{\left(\log{\left(x \right)} \sin{\left(\sqrt{x} \right)} + \tan{\left(4 x \right)} \right)}$$

x*log(sin(sqrt(x))*log(x) + tan(4*x))

Detail solution

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 $1$
$g{\left(x \right)} = \log{\left(\log{\left(x \right)} \sin{\left(\sqrt{x} \right)} + \tan{\left(4 x \right)} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)} \sin{\left(\sqrt{x} \right)} + \tan{\left(4 x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\log{\left(x \right)} \sin{\left(\sqrt{x} \right)} + \tan{\left(4 x \right)}\right)$ :
  1. Differentiate $\log{\left(x \right)} \sin{\left(\sqrt{x} \right)} + \tan{\left(4 x \right)}$ term by term:
    1. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = \sin{\left(\sqrt{x} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Let $u = \sqrt{x}$ .
      2. The derivative of sine is cosine:
        
        $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      3. 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:
        
        $\frac{\cos{\left(\sqrt{x} \right)}}{2 \sqrt{x}}$
      $g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      The result is: $\frac{\sin{\left(\sqrt{x} \right)}}{x} + \frac{\log{\left(x \right)} \cos{\left(\sqrt{x} \right)}}{2 \sqrt{x}}$
    2. Rewrite the function to be differentiated:
      
      $\tan{\left(4 x \right)} = \frac{\sin{\left(4 x \right)}}{\cos{\left(4 x \right)}}$
    3. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(4 x \right)}$ and $g{\left(x \right)} = \cos{\left(4 x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Let $u = 4 x$ .
      2. The derivative of sine is cosine:
        
        $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$ :
        
        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: $4$
        
        The result of the chain rule is:
        
        $4 \cos{\left(4 x \right)}$
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      1. Let $u = 4 x$ .
      2. The derivative of cosine is negative sine:
        
        $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$ :
        
        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: $4$
        
        The result of the chain rule is:
        
        $- 4 \sin{\left(4 x \right)}$
      Now plug in to the quotient rule:
      
      $\frac{4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}}$
    The result is: $\frac{4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}} + \frac{\sin{\left(\sqrt{x} \right)}}{x} + \frac{\log{\left(x \right)} \cos{\left(\sqrt{x} \right)}}{2 \sqrt{x}}$
  The result of the chain rule is:
  
  $\frac{\frac{4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}} + \frac{\sin{\left(\sqrt{x} \right)}}{x} + \frac{\log{\left(x \right)} \cos{\left(\sqrt{x} \right)}}{2 \sqrt{x}}}{\log{\left(x \right)} \sin{\left(\sqrt{x} \right)} + \tan{\left(4 x \right)}}$
The result is: $\frac{x \left(\frac{4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}} + \frac{\sin{\left(\sqrt{x} \right)}}{x} + \frac{\log{\left(x \right)} \cos{\left(\sqrt{x} \right)}}{2 \sqrt{x}}\right)}{\log{\left(x \right)} \sin{\left(\sqrt{x} \right)} + \tan{\left(4 x \right)}} + \log{\left(\log{\left(x \right)} \sin{\left(\sqrt{x} \right)} + \tan{\left(4 x \right)} \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  /                     /  ___\      /  ___\       \                                    
  |         2        sin\\/ x /   cos\\/ x /*log(x)|                                    
x*|4 + 4*tan (4*x) + ---------- + -----------------|                                    
  |                      x                 ___     |                                    
  \                                    2*\/ x      /      /   /  ___\                  \
---------------------------------------------------- + log\sin\\/ x /*log(x) + tan(4*x)/
               /  ___\                                                                  
            sin\\/ x /*log(x) + tan(4*x)

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

Simplify

The second derivative [src]

                                   /                                                    2                                                                                                     \                    
                                   |/                       /  ___\      /  ___\       \                                                                                                      |                    
                                   ||         2        2*sin\\/ x /   cos\\/ x /*log(x)|                                                                                                      |                    
                                   ||8 + 8*tan (4*x) + ------------ + -----------------|                                                                                                      |                    
                                   ||                       x                 ___      |                                        /  ___\        /  ___\             /  ___\      /  ___\       |                    
                                   |\                                       \/ x       /        /       2     \            4*cos\\/ x /   4*sin\\/ x /   log(x)*sin\\/ x /   cos\\/ x /*log(x)|                    
                                 x*|----------------------------------------------------- - 128*\1 + tan (4*x)/*tan(4*x) - ------------ + ------------ + ----------------- + -----------------|                    
                       /  ___\     |                       /  ___\                                                              3/2             2                x                   3/2      |      /  ___\       
         2        2*sin\\/ x /     \             log(x)*sin\\/ x / + tan(4*x)                                                  x               x                                    x         /   cos\\/ x /*log(x)
8 + 8*tan (4*x) + ------------ - -------------------------------------------------------------------------------------------------------------------------------------------------------------- + -----------------
                       x                                                                                       4                                                                                          ___      
                                                                                                                                                                                                        \/ x       
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                      /  ___\                                                                                                      
                                                                                            log(x)*sin\\/ x / + tan(4*x)

$$\frac{- \frac{x \left(- 128 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 x \right)} + \frac{\left(8 \tan^{2}{\left(4 x \right)} + 8 + \frac{2 \sin{\left(\sqrt{x} \right)}}{x} + \frac{\log{\left(x \right)} \cos{\left(\sqrt{x} \right)}}{\sqrt{x}}\right)^{2}}{\log{\left(x \right)} \sin{\left(\sqrt{x} \right)} + \tan{\left(4 x \right)}} + \frac{\log{\left(x \right)} \sin{\left(\sqrt{x} \right)}}{x} + \frac{4 \sin{\left(\sqrt{x} \right)}}{x^{2}} + \frac{\log{\left(x \right)} \cos{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}}} - \frac{4 \cos{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}}}\right)}{4} + 8 \tan^{2}{\left(4 x \right)} + 8 + \frac{2 \sin{\left(\sqrt{x} \right)}}{x} + \frac{\log{\left(x \right)} \cos{\left(\sqrt{x} \right)}}{\sqrt{x}}}{\log{\left(x \right)} \sin{\left(\sqrt{x} \right)} + \tan{\left(4 x \right)}}$$

Simplify

The third derivative [src]

                                                                                                                          /                                                                                                             3                                                                                                                                                                                                                                                                                 \                                            
                                                                                                                          |                                                         /                       /  ___\      /  ___\       \                                                                                                                       /                       /  ___\      /  ___\       \ /                                      /  ___\        /  ___\             /  ___\      /  ___\       \|                                            
                                                                                                                          |                                                         |         2        2*sin\\/ x /   cos\\/ x /*log(x)|                                                                                                                       |         2        2*sin\\/ x /   cos\\/ x /*log(x)| |      /       2     \            4*cos\\/ x /   4*sin\\/ x /   log(x)*sin\\/ x /   cos\\/ x /*log(x)||                                            
                                                                                                                          |                                                       2*|8 + 8*tan (4*x) + ------------ + -----------------|                                                                                                                     3*|8 + 8*tan (4*x) + ------------ + -----------------|*|- 128*\1 + tan (4*x)/*tan(4*x) - ------------ + ------------ + ----------------- + -----------------||                                            
                                                                                                                    2     |                    2         /  ___\        /  ___\     |                       x                 ___      |          /  ___\                                       /  ___\                      /  ___\        /  ___\            |                       x                 ___      | |                                      3/2             2                x                   3/2      ||                                            
                                                                /                       /  ___\      /  ___\       \      |     /       2     \    18*cos\\/ x /   6*sin\\/ x /     \                                       \/ x       /    16*sin\\/ x /           2      /       2     \   cos\\/ x /*log(x)   3*log(x)*sin\\/ x /   3*cos\\/ x /*log(x)     \                                       \/ x       / \                                     x               x                                    x         /|                                            
                                                                |         2        2*sin\\/ x /   cos\\/ x /*log(x)|    x*|1024*\1 + tan (4*x)/  - ------------- - ------------ + ------------------------------------------------------- + ------------- + 2048*tan (4*x)*\1 + tan (4*x)/ - ----------------- + ------------------- + ------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------|                                            
                                                              3*|8 + 8*tan (4*x) + ------------ + -----------------|      |                              5/2             2                                                  2                      3                                                 3/2                   2                    5/2                                                                                     /  ___\                                                                           |                                            
       /  ___\        /  ___\                                   |                       x                 ___      |      |                             x               x                     /          /  ___\           \                      x                                                 x                     x                    x                                                                              log(x)*sin\\/ x / + tan(4*x)                                                                |               /  ___\        /  ___\       
  3*sin\\/ x /   3*cos\\/ x /      /       2     \              \                                       \/ x       /      \                                                                   \log(x)*sin\\/ x / + tan(4*x)/                                                                                                                                                                                                                                                                                              /   3*log(x)*sin\\/ x /   3*cos\\/ x /*log(x)
- ------------ + ------------ + 96*\1 + tan (4*x)/*tan(4*x) - ------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------- - -------------------
        2             3/2                                                   /          /  ___\           \                                                                                                                                                                                                               8                                                                                                                                                                                                            4*x                     3/2      
       x             x                                                    4*\log(x)*sin\\/ x / + tan(4*x)/                                                                                                                                                                                                                                                                                                                                                                                                                                                 4*x         
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                /  ___\                                                                                                                                                                                                                                                                                
                                                                                                                                                                                                                                                                      log(x)*sin\\/ x / + tan(4*x)

$$\frac{\frac{x \left(1024 \left(\tan^{2}{\left(4 x \right)} + 1\right)^{2} + 2048 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan^{2}{\left(4 x \right)} + \frac{3 \left(8 \tan^{2}{\left(4 x \right)} + 8 + \frac{2 \sin{\left(\sqrt{x} \right)}}{x} + \frac{\log{\left(x \right)} \cos{\left(\sqrt{x} \right)}}{\sqrt{x}}\right) \left(- 128 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 x \right)} + \frac{\log{\left(x \right)} \sin{\left(\sqrt{x} \right)}}{x} + \frac{4 \sin{\left(\sqrt{x} \right)}}{x^{2}} + \frac{\log{\left(x \right)} \cos{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}}} - \frac{4 \cos{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}}}\right)}{\log{\left(x \right)} \sin{\left(\sqrt{x} \right)} + \tan{\left(4 x \right)}} + \frac{2 \left(8 \tan^{2}{\left(4 x \right)} + 8 + \frac{2 \sin{\left(\sqrt{x} \right)}}{x} + \frac{\log{\left(x \right)} \cos{\left(\sqrt{x} \right)}}{\sqrt{x}}\right)^{3}}{\left(\log{\left(x \right)} \sin{\left(\sqrt{x} \right)} + \tan{\left(4 x \right)}\right)^{2}} + \frac{3 \log{\left(x \right)} \sin{\left(\sqrt{x} \right)}}{x^{2}} - \frac{6 \sin{\left(\sqrt{x} \right)}}{x^{2}} + \frac{16 \sin{\left(\sqrt{x} \right)}}{x^{3}} - \frac{\log{\left(x \right)} \cos{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}}} + \frac{3 \log{\left(x \right)} \cos{\left(\sqrt{x} \right)}}{x^{\frac{5}{2}}} - \frac{18 \cos{\left(\sqrt{x} \right)}}{x^{\frac{5}{2}}}\right)}{8} + 96 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 x \right)} - \frac{3 \left(8 \tan^{2}{\left(4 x \right)} + 8 + \frac{2 \sin{\left(\sqrt{x} \right)}}{x} + \frac{\log{\left(x \right)} \cos{\left(\sqrt{x} \right)}}{\sqrt{x}}\right)^{2}}{4 \left(\log{\left(x \right)} \sin{\left(\sqrt{x} \right)} + \tan{\left(4 x \right)}\right)} - \frac{3 \log{\left(x \right)} \sin{\left(\sqrt{x} \right)}}{4 x} - \frac{3 \sin{\left(\sqrt{x} \right)}}{x^{2}} - \frac{3 \log{\left(x \right)} \cos{\left(\sqrt{x} \right)}}{4 x^{\frac{3}{2}}} + \frac{3 \cos{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}}}}{\log{\left(x \right)} \sin{\left(\sqrt{x} \right)} + \tan{\left(4 x \right)}}$$

Simplify

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Derivative of y=xln(sinsqrtxlnx+tg4x)

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