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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   /   2       \
log\sin (x) + 1/
$$\log{\left(\sin^{2}{\left(x \right)} + 1 \right)}$$
log(sin(x)^2 + 1)
Detail solution
  1. Let .

  2. The derivative of is .

  3. Then, apply the chain rule. Multiply by :

    1. Differentiate term by term:

      1. Let .

      2. Apply the power rule: goes to

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of sine is cosine:

        The result of the chain rule is:

      4. The derivative of the constant is zero.

      The result is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
2*cos(x)*sin(x)
---------------
     2         
  sin (x) + 1  
$$\frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}$$
The second derivative [src]
  /                         2       2   \
  |   2         2      2*cos (x)*sin (x)|
2*|cos (x) - sin (x) - -----------------|
  |                              2      |
  \                       1 + sin (x)   /
-----------------------------------------
                      2                  
               1 + sin (x)               
$$\frac{2 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} - \frac{2 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}\right)}{\sin^{2}{\left(x \right)} + 1}$$
The third derivative [src]
  /           2             2            2       2   \              
  |      3*cos (x)     3*sin (x)    4*cos (x)*sin (x)|              
4*|-2 - ----------- + ----------- + -----------------|*cos(x)*sin(x)
  |            2             2                     2 |              
  |     1 + sin (x)   1 + sin (x)     /       2   \  |              
  \                                   \1 + sin (x)/  /              
--------------------------------------------------------------------
                                   2                                
                            1 + sin (x)                             
$$\frac{4 \left(-2 + \frac{3 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} - \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{4 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1}$$