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The derivative of the function/ y=ln(2x/sin(x^-1))

Derivative of y=ln(2x/sin(x^-1))

The solution

You have entered [src]

   / 2*x  \
log|------|
   |   /1\|
   |sin|-||
   \   \x//

$$\log{\left(\frac{2 x}{\sin{\left(\frac{1}{x} \right)}} \right)}$$

log((2*x)/sin(1/x))

Detail solution

Let $u = \frac{2 x}{\sin{\left(\frac{1}{x} \right)}}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{2 x}{\sin{\left(\frac{1}{x} \right)}}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 2 x$ and $g{\left(x \right)} = \sin{\left(\frac{1}{x} \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $2$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \frac{1}{x}$ .
  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 $\frac{d}{d x} \frac{1}{x}$ :
    1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
    The result of the chain rule is:
    
    $- \frac{\cos{\left(\frac{1}{x} \right)}}{x^{2}}$
  Now plug in to the quotient rule:
  
  $\frac{2 \sin{\left(\frac{1}{x} \right)} + \frac{2 \cos{\left(\frac{1}{x} \right)}}{x}}{\sin^{2}{\left(\frac{1}{x} \right)}}$
The result of the chain rule is:

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

$\frac{x + \frac{1}{\tan{\left(\frac{1}{x} \right)}}}{x^{2}}$

The answer is:

$\frac{x + \frac{1}{\tan{\left(\frac{1}{x} \right)}}}{x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/               /1\\       
|          2*cos|-||       
|  2            \x/|    /1\
|------ + ---------|*sin|-|
|   /1\        2/1\|    \x/
|sin|-|   x*sin |-||       
\   \x/         \x//       
---------------------------
            2*x

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

Simplify

The second derivative [src]

              2/1\              /        /1\ \       
         2*cos |-|              |     cos|-| |       
               \x/              |        \x/ |    /1\
     1 + ---------              |1 + --------|*cos|-|
             2/1\        /1\    |         /1\|    \x/
          sin |-|     cos|-|    |    x*sin|-||       
              \x/        \x/    \         \x//       
-1 + ------------- - -------- - ---------------------
            2             /1\               /1\      
           x         x*sin|-|          x*sin|-|      
                          \x/               \x/      
-----------------------------------------------------
                           2                         
                          x

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

Simplify

The third derivative [src]

            /1\             2/1\        3/1\        /1\     /         2/1\\                /         2/1\\            /        /1\ \       
         cos|-|        6*cos |-|   6*cos |-|   5*cos|-|     |    2*cos |-||                |    2*cos |-||            |     cos|-| |       
            \x/              \x/         \x/        \x/     |          \x/|                |          \x/|    /1\     |        \x/ |    /1\
    1 + --------   3 + --------- - --------- - --------   2*|1 + ---------|              2*|1 + ---------|*cos|-|   4*|1 + --------|*cos|-|
             /1\           2/1\         3/1\        /1\     |        2/1\ |        /1\     |        2/1\ |    \x/     |         /1\|    \x/
        x*sin|-|        sin |-|    x*sin |-|   x*sin|-|     |     sin |-| |   2*cos|-|     |     sin |-| |            |    x*sin|-||       
             \x/            \x/          \x/        \x/     \         \x/ /        \x/     \         \x/ /            \         \x//       
2 - ------------ - ------------------------------------ - ----------------- + -------- - ------------------------ + -----------------------
          2                          2                             2               /1\           3    /1\                        /1\       
         x                          x                             x           x*sin|-|          x *sin|-|                   x*sin|-|       
                                                                                   \x/                \x/                        \x/       
-------------------------------------------------------------------------------------------------------------------------------------------
                                                                      3                                                                    
                                                                     x

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

Simplify

The graph

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Derivative of y=ln(2x/sin(x^-1))

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