Derivative of ln(1+sinx/cosx)

The solution

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

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

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

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

Transform

Detail solution

Let $u = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1\right)$ :
1. Differentiate $\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. 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 $g{\left(x \right)} = \cos{\left(x \right)}$ .
    
    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 cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  The result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result of the chain rule is:

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

$\frac{2 \sqrt{2}}{2 \sin{\left(2 x + \frac{\pi}{4} \right)} + \sqrt{2}}$

The answer is:

$\frac{2 \sqrt{2}}{2 \sin{\left(2 x + \frac{\pi}{4} \right)} + \sqrt{2}}$

The graph

Plot the graph f(x)

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ln(1+sinx/cosx)

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