Mister Exam

Derivative of sinx/(1+tanx)

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

The graph:

from to

Enter:

The solution

You have entered [src]
  sin(x)  
----------
1 + tan(x)
$$\frac{\sin{\left(x \right)}}{\tan{\left(x \right)} + 1}$$
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. The derivative of sine is cosine:

    To find :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. Rewrite the function to be differentiated:

      3. Apply the quotient rule, which is:

        and .

        To find :

        1. The derivative of sine is cosine:

        To find :

        1. The derivative of cosine is negative sine:

        Now plug in to the quotient rule:

      The result is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
             /        2   \       
  cos(x)     \-1 - tan (x)/*sin(x)
---------- + ---------------------
1 + tan(x)                   2    
                 (1 + tan(x))     
$$\frac{\cos{\left(x \right)}}{\tan{\left(x \right)} + 1} + \frac{\left(- \tan^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}}$$
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)                                     
$$- \frac{\sin{\left(x \right)} + \frac{2 \left(\tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} + 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\tan{\left(x \right)} + 1} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\tan{\left(x \right)} + 1}}{\tan{\left(x \right)} + 1}$$
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)                                                                              
$$\frac{- \cos{\left(x \right)} - \frac{6 \left(\tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} + 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\tan{\left(x \right)} + 1} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1 - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{\tan{\left(x \right)} + 1} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\left(\tan{\left(x \right)} + 1\right)^{2}}\right) \sin{\left(x \right)}}{\tan{\left(x \right)} + 1} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\tan{\left(x \right)} + 1}}{\tan{\left(x \right)} + 1}$$
The graph
Derivative of sinx/(1+tanx)
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