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The derivative of the function/ (sin(x))/log(x^2+1)

Derivative of (sin(x))/log(x^2+1)

The solution

You have entered [src]

   sin(x)  
-----------
   / 2    \
log\x  + 1/

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

d /   sin(x)  \
--|-----------|
dx|   / 2    \|
  \log\x  + 1//

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

Transform

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 $g{\left(x \right)} = \log{\left(x^{2} + 1 \right)}$ .

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. Let $u = x^{2} + 1$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 1\right)$ :
  1. Differentiate $x^{2} + 1$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    2. The derivative of the constant $1$ is zero.
    The result is: $2 x$
  The result of the chain rule is:
  
  $\frac{2 x}{x^{2} + 1}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                                   /         2               2        \       
                                   |      2*x             4*x         |       
                                 2*|-1 + ------ + --------------------|*sin(x)
                                   |          2   /     2\    /     2\|       
               4*x*cos(x)          \     1 + x    \1 + x /*log\1 + x //       
-sin(x) - -------------------- + ---------------------------------------------
          /     2\    /     2\                /     2\    /     2\            
          \1 + x /*log\1 + x /                \1 + x /*log\1 + x /            
------------------------------------------------------------------------------
                                    /     2\                                  
                                 log\1 + x /

$$\frac{- \frac{4 x \cos{\left(x \right)}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}} - \sin{\left(x \right)} + \frac{2 \cdot \left(\frac{2 x^{2}}{x^{2} + 1} + \frac{4 x^{2}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}} - 1\right) \sin{\left(x \right)}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}}}{\log{\left(x^{2} + 1 \right)}}$$

Simplify

The third derivative [src]

                                   /         2               2        \              /                       2               2                       2        \       
                                   |      2*x             4*x         |              |          6         4*x            12*x                    12*x         |       
                                 6*|-1 + ------ + --------------------|*cos(x)   4*x*|-3 - ----------- + ------ + -------------------- + ---------------------|*sin(x)
                                   |          2   /     2\    /     2\|              |        /     2\        2   /     2\    /     2\   /     2\    2/     2\|       
               6*x*sin(x)          \     1 + x    \1 + x /*log\1 + x //              \     log\1 + x /   1 + x    \1 + x /*log\1 + x /   \1 + x /*log \1 + x //       
-cos(x) + -------------------- + --------------------------------------------- - -------------------------------------------------------------------------------------
          /     2\    /     2\                /     2\    /     2\                                                       2                                            
          \1 + x /*log\1 + x /                \1 + x /*log\1 + x /                                               /     2\     /     2\                                
                                                                                                                 \1 + x / *log\1 + x /                                
----------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                /     2\                                                                              
                                                                             log\1 + x /

$$\frac{\frac{6 x \sin{\left(x \right)}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}} - \frac{4 x \left(\frac{4 x^{2}}{x^{2} + 1} + \frac{12 x^{2}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}} + \frac{12 x^{2}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}^{2}} - 3 - \frac{6}{\log{\left(x^{2} + 1 \right)}}\right) \sin{\left(x \right)}}{\left(x^{2} + 1\right)^{2} \log{\left(x^{2} + 1 \right)}} - \cos{\left(x \right)} + \frac{6 \cdot \left(\frac{2 x^{2}}{x^{2} + 1} + \frac{4 x^{2}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}} - 1\right) \cos{\left(x \right)}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}}}{\log{\left(x^{2} + 1 \right)}}$$

Simplify

The graph

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Derivative of (sin(x))/log(x^2+1)

The graph:

Piecewise:

The solution