Mister Exam

# Derivative of sinx/(1+tanx)

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

from to

### The solution

You have entered [src]
  sin(x)
----------
1 + tan(x)
$$\frac{\sin{\left(x \right)}}{\tan{\left(x \right)} + 1}$$
d /  sin(x)  \
--|----------|
dx\1 + tan(x)/
$$\frac{d}{d x} \frac{\sin{\left(x \right)}}{\tan{\left(x \right)} + 1}$$
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 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)                                     
$$- \frac{\frac{2 \left(\tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} + 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\tan{\left(x \right)} + 1} + \sin{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\tan{\left(x \right)} + 1}}{\tan{\left(x \right)} + 1}$$
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)                                                                              
$$\frac{- \frac{6 \left(\tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} + 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\tan{\left(x \right)} + 1} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{\tan{\left(x \right)} + 1} + 1 + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\left(\tan{\left(x \right)} + 1\right)^{2}}\right) \sin{\left(x \right)}}{\tan{\left(x \right)} + 1} - \cos{\left(x \right)} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\tan{\left(x \right)} + 1}}{\tan{\left(x \right)} + 1}$$
The graph