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sin(x)/(log(x)+ one)
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sin(x)/(log(x)-1)
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The derivative of the function/ sin(x)/(log(x)+1)

Derivative of sin(x)/(log(x)+1)

The solution

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  sin(x)  
----------
log(x) + 1

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

sin(x)/(log(x) + 1)

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 $g{\left(x \right)} = \log{\left(x \right)} + 1$ .

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. Differentiate $\log{\left(x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $\frac{1}{x}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of sin(x)/(log(x)+1)

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The solution