Derivative of (sinx)/(lnx)

The solution

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sin(x)
------
log(x)

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

sin(x)/log(x)

Detail solution

Apply the quotient rule, which is:

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

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

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 $\log{\left(x \right)}$ is $\frac{1}{x}$ .
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

cos(x)     sin(x) 
------ - ---------
log(x)        2   
         x*log (x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (sinx)/(lnx)

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The solution