Derivative of secxcosx/log(tanx)

The solution

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sec(x)*cos(x)
-------------
 log(tan(x))

$$\frac{\cos{\left(x \right)} \sec{\left(x \right)}}{\log{\left(\tan{\left(x \right)} \right)}}$$

(sec(x)*cos(x))/log(tan(x))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \cos{\left(x \right)} \sec{\left(x \right)}$ and $g{\left(x \right)} = \log{\left(\tan{\left(x \right)} \right)}$ .

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

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Derivative of secxcosx/log(tanx)

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