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

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

The solution

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

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

log(sin(x)^2 + 1)

Detail solution

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

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

$\frac{2 \sin{\left(2 x \right)}}{3 - \cos{\left(2 x \right)}}$

The answer is:

$\frac{2 \sin{\left(2 x \right)}}{3 - \cos{\left(2 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

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)

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

Simplify

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)

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

Simplify

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

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The solution