Derivative of ln(x)/sin(x)+x*ctg(x)

The solution

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log(x)           
------ + x*cot(x)
sin(x)

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

log(x)/sin(x) + x*cot(x)

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                                                                                  2          
          2      log(x)       1        2*cos(x)       /       2   \          2*cos (x)*log(x)
-2 - 2*cot (x) + ------ - --------- - --------- + 2*x*\1 + cot (x)/*cot(x) + ----------------
                 sin(x)    2               2                                        3        
                          x *sin(x)   x*sin (x)                                  sin (x)

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

Simplify

The third derivative [src]

                   2                                                        3                                                                             2   
      /       2   \        2          3         /       2   \          6*cos (x)*log(x)   5*cos(x)*log(x)          2    /       2   \    3*cos(x)    6*cos (x)
- 2*x*\1 + cot (x)/  + --------- + -------- + 6*\1 + cot (x)/*cot(x) - ---------------- - --------------- - 4*x*cot (x)*\1 + cot (x)/ + ---------- + ---------
                        3          x*sin(x)                                   4                  2                                       2    2           3   
                       x *sin(x)                                           sin (x)            sin (x)                                   x *sin (x)   x*sin (x)

$$- 2 x \left(\cot^{2}{\left(x \right)} + 1\right)^{2} - 4 x \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} + 6 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \frac{5 \log{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{6 \log{\left(x \right)} \cos^{3}{\left(x \right)}}{\sin^{4}{\left(x \right)}} + \frac{3}{x \sin{\left(x \right)}} + \frac{6 \cos^{2}{\left(x \right)}}{x \sin^{3}{\left(x \right)}} + \frac{3 \cos{\left(x \right)}}{x^{2} \sin^{2}{\left(x \right)}} + \frac{2}{x^{3} \sin{\left(x \right)}}$$

Simplify

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Derivative of ln(x)/sin(x)+x*ctg(x)

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Piecewise:

The solution

Method #1

Method #2