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Identical expressions
(tg(logx))/(sin(logx)+ five)
(tg( logarithm of x)) divide by ( sinus of ( logarithm of x) plus 5)
(tg( logarithm of x)) divide by ( sinus of ( logarithm of x) plus five)
tglogx/sinlogx+5
(tg(logx)) divide by (sin(logx)+5)
Similar expressions
(tg(logx))/(sin(logx)-5)

The derivative of the function/ (tg(logx))/(sin(logx)+5)

Derivative of (tg(logx))/(sin(logx)+5)

The solution

You have entered [src]

  tan(log(x))  
---------------
sin(log(x)) + 5

$$\frac{\tan{\left(\log{\left(x \right)} \right)}}{\sin{\left(\log{\left(x \right)} \right)} + 5}$$

tan(log(x))/(sin(log(x)) + 5)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \tan{\left(\log{\left(x \right)} \right)}$ and $g{\left(x \right)} = \sin{\left(\log{\left(x \right)} \right)} + 5$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(\log{\left(x \right)} \right)} = \frac{\sin{\left(\log{\left(x \right)} \right)}}{\cos{\left(\log{\left(x \right)} \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(\log{\left(x \right)} \right)}$ and $g{\left(x \right)} = \cos{\left(\log{\left(x \right)} \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = \log{\left(x \right)}$ .
  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 $\frac{d}{d x} \log{\left(x \right)}$ :
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    The result of the chain rule is:
    
    $\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \log{\left(x \right)}$ .
  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 $\frac{d}{d x} \log{\left(x \right)}$ :
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    The result of the chain rule is:
    
    $- \frac{\sin{\left(\log{\left(x \right)} \right)}}{x}$
  Now plug in to the quotient rule:
  
  $\frac{\frac{\sin^{2}{\left(\log{\left(x \right)} \right)}}{x} + \frac{\cos^{2}{\left(\log{\left(x \right)} \right)}}{x}}{\cos^{2}{\left(\log{\left(x \right)} \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\sin{\left(\log{\left(x \right)} \right)} + 5$ term by term:
  1. The derivative of the constant $5$ is zero.
  2. Let $u = \log{\left(x \right)}$ .
  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 $\frac{d}{d x} \log{\left(x \right)}$ :
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    The result of the chain rule is:
    
    $\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$
  The result is: $\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$
Now plug in to the quotient rule:

$\frac{\frac{\left(\frac{\sin^{2}{\left(\log{\left(x \right)} \right)}}{x} + \frac{\cos^{2}{\left(\log{\left(x \right)} \right)}}{x}\right) \left(\sin{\left(\log{\left(x \right)} \right)} + 5\right)}{\cos^{2}{\left(\log{\left(x \right)} \right)}} - \frac{\cos{\left(\log{\left(x \right)} \right)} \tan{\left(\log{\left(x \right)} \right)}}{x}}{\left(\sin{\left(\log{\left(x \right)} \right)} + 5\right)^{2}}$
Now simplify:

$\frac{\sin^{3}{\left(\log{\left(x \right)} \right)} + 5}{x \left(\sin{\left(\log{\left(x \right)} \right)} + 5\right)^{2} \cos^{2}{\left(\log{\left(x \right)} \right)}}$

The answer is:

$\frac{\sin^{3}{\left(\log{\left(x \right)} \right)} + 5}{x \left(\sin{\left(\log{\left(x \right)} \right)} + 5\right)^{2} \cos^{2}{\left(\log{\left(x \right)} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         2                                   
  1 + tan (log(x))    cos(log(x))*tan(log(x))
------------------- - -----------------------
x*(sin(log(x)) + 5)                        2 
                        x*(sin(log(x)) + 5)

$$\frac{\tan^{2}{\left(\log{\left(x \right)} \right)} + 1}{x \left(\sin{\left(\log{\left(x \right)} \right)} + 5\right)} - \frac{\cos{\left(\log{\left(x \right)} \right)} \tan{\left(\log{\left(x \right)} \right)}}{x \left(\sin{\left(\log{\left(x \right)} \right)} + 5\right)^{2}}$$

Simplify

The second derivative [src]

                                          /      2                                    \                                               
                                          | 2*cos (log(x))                            |                                               
                                          |--------------- + cos(log(x)) + sin(log(x))|*tan(log(x))     /       2        \            
/       2        \                        \5 + sin(log(x))                            /               2*\1 + tan (log(x))/*cos(log(x))
\1 + tan (log(x))/*(-1 + 2*tan(log(x))) + --------------------------------------------------------- - --------------------------------
                                                               5 + sin(log(x))                                5 + sin(log(x))         
--------------------------------------------------------------------------------------------------------------------------------------
                                                          2                                                                           
                                                         x *(5 + sin(log(x)))

$$\frac{\left(2 \tan{\left(\log{\left(x \right)} \right)} - 1\right) \left(\tan^{2}{\left(\log{\left(x \right)} \right)} + 1\right) - \frac{2 \left(\tan^{2}{\left(\log{\left(x \right)} \right)} + 1\right) \cos{\left(\log{\left(x \right)} \right)}}{\sin{\left(\log{\left(x \right)} \right)} + 5} + \frac{\left(\sin{\left(\log{\left(x \right)} \right)} + \cos{\left(\log{\left(x \right)} \right)} + \frac{2 \cos^{2}{\left(\log{\left(x \right)} \right)}}{\sin{\left(\log{\left(x \right)} \right)} + 5}\right) \tan{\left(\log{\left(x \right)} \right)}}{\sin{\left(\log{\left(x \right)} \right)} + 5}}{x^{2} \left(\sin{\left(\log{\left(x \right)} \right)} + 5\right)}$$

Simplify

The third derivative [src]

                                                            /                      2                  3                                                    \                                                                                                                                         
                                                            |                 6*cos (log(x))     6*cos (log(x))     6*cos(log(x))*sin(log(x))              |                                    /      2                                    \                                                        
                                                            |3*sin(log(x)) + --------------- + ------------------ + ------------------------- + cos(log(x))|*tan(log(x))     /       2        \ | 2*cos (log(x))                            |                                                        
                                                            |                5 + sin(log(x))                    2        5 + sin(log(x))                   |               3*\1 + tan (log(x))/*|--------------- + cos(log(x)) + sin(log(x))|     /       2        \                                 
  /       2        \ /                         2        \   \                                  (5 + sin(log(x)))                                           /                                    \5 + sin(log(x))                            /   3*\1 + tan (log(x))/*(-1 + 2*tan(log(x)))*cos(log(x))
2*\1 + tan (log(x))/*\2 - 3*tan(log(x)) + 3*tan (log(x))/ - ------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------ - -----------------------------------------------------
                                                                                                          5 + sin(log(x))                                                                           5 + sin(log(x))                                                5 + sin(log(x))                   
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                          3                                                                                                                                                          
                                                                                                                                         x *(5 + sin(log(x)))

$$\frac{2 \left(\tan^{2}{\left(\log{\left(x \right)} \right)} + 1\right) \left(3 \tan^{2}{\left(\log{\left(x \right)} \right)} - 3 \tan{\left(\log{\left(x \right)} \right)} + 2\right) - \frac{3 \left(2 \tan{\left(\log{\left(x \right)} \right)} - 1\right) \left(\tan^{2}{\left(\log{\left(x \right)} \right)} + 1\right) \cos{\left(\log{\left(x \right)} \right)}}{\sin{\left(\log{\left(x \right)} \right)} + 5} + \frac{3 \left(\tan^{2}{\left(\log{\left(x \right)} \right)} + 1\right) \left(\sin{\left(\log{\left(x \right)} \right)} + \cos{\left(\log{\left(x \right)} \right)} + \frac{2 \cos^{2}{\left(\log{\left(x \right)} \right)}}{\sin{\left(\log{\left(x \right)} \right)} + 5}\right)}{\sin{\left(\log{\left(x \right)} \right)} + 5} - \frac{\left(3 \sin{\left(\log{\left(x \right)} \right)} + \cos{\left(\log{\left(x \right)} \right)} + \frac{6 \sin{\left(\log{\left(x \right)} \right)} \cos{\left(\log{\left(x \right)} \right)}}{\sin{\left(\log{\left(x \right)} \right)} + 5} + \frac{6 \cos^{2}{\left(\log{\left(x \right)} \right)}}{\sin{\left(\log{\left(x \right)} \right)} + 5} + \frac{6 \cos^{3}{\left(\log{\left(x \right)} \right)}}{\left(\sin{\left(\log{\left(x \right)} \right)} + 5\right)^{2}}\right) \tan{\left(\log{\left(x \right)} \right)}}{\sin{\left(\log{\left(x \right)} \right)} + 5}}{x^{3} \left(\sin{\left(\log{\left(x \right)} \right)} + 5\right)}$$

Simplify

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Derivative of (tg(logx))/(sin(logx)+5)

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