Derivative of sinx/(1+tanx)

The solution

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  sin(x)  
----------
1 + tan(x)

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

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

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

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 $g{\left(x \right)} = \tan{\left(x \right)} + 1$ .

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. Differentiate $\tan{\left(x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Rewrite the function to be differentiated:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  3. 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: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of sinx/(1+tanx)

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The solution