Derivative of ln*((1+sinx)/cosx)

The solution

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

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

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

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                                                                               /                    2                \            /                    2                \                                                                                         
                                                                                                               |               2*sin (x)*(1 + sin(x))|            |               2*sin (x)*(1 + sin(x))|               3    /    (1 + sin(x))*sin(x)\     /    (1 + sin(x))*sin(x)\              
                                                                                                             2*|1 + 2*sin(x) + ----------------------|*cos(x)   2*|1 + 2*sin(x) + ----------------------|*sin(x)   2*cos (x)*|1 + -------------------|   3*|1 + -------------------|*cos(x)*sin(x)
/         2                                   3                \                                               |                         2           |            |                         2           |                    |             2         |     |             2         |              
|    3*sin (x)   5*(1 + sin(x))*sin(x)   6*sin (x)*(1 + sin(x))|          /    (1 + sin(x))*sin(x)\            \                      cos (x)        /            \                      cos (x)        /                    \          cos (x)      /     \          cos (x)      /              
|2 + --------- + --------------------- + ----------------------|*cos(x) - |1 + -------------------|*cos(x) - ------------------------------------------------ - ------------------------------------------------ + ----------------------------------- + -----------------------------------------
|        2                 2                       4           |          |             2         |                             1 + sin(x)                                           cos(x)                                               2                              1 + sin(x)               
\     cos (x)           cos (x)                 cos (x)        /          \          cos (x)      /                                                                                                                           (1 + sin(x))                                                        
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                            1 + sin(x)

$$\frac{- \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cos{\left(x \right)} + \frac{3 \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)} + 1} + \frac{2 \left(\frac{\left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cos^{3}{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} - \frac{2 \left(\frac{2 \left(\sin{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 \sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \left(\frac{6 \left(\sin{\left(x \right)} + 1\right) \sin^{3}{\left(x \right)}}{\cos^{4}{\left(x \right)}} + \frac{5 \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) \cos{\left(x \right)} - \frac{2 \left(\frac{2 \left(\sin{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2 \sin{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)} + 1}}{\sin{\left(x \right)} + 1}$$

Simplify

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Derivative of ln*((1+sinx)/cosx)

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