Mister Exam

Derivative of tan(x)/(x+2)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
tan(x)
------
x + 2 
$$\frac{\tan{\left(x \right)}}{x + 2}$$
tan(x)/(x + 2)
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Rewrite the function to be differentiated:

    2. 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:

    To find :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. Apply the power rule: goes to

      The result is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
       2              
1 + tan (x)    tan(x) 
----------- - --------
   x + 2             2
              (x + 2) 
$$\frac{\tan^{2}{\left(x \right)} + 1}{x + 2} - \frac{\tan{\left(x \right)}}{\left(x + 2\right)^{2}}$$
The second derivative [src]
  /                                         2   \
  | tan(x)    /       2   \          1 + tan (x)|
2*|-------- + \1 + tan (x)/*tan(x) - -----------|
  |       2                             2 + x   |
  \(2 + x)                                      /
-------------------------------------------------
                      2 + x                      
$$\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{x + 2} + \frac{\tan{\left(x \right)}}{\left(x + 2\right)^{2}}\right)}{x + 2}$$
The third derivative [src]
  /                                             /       2   \     /       2   \       \
  |/       2   \ /         2   \   3*tan(x)   3*\1 + tan (x)/   3*\1 + tan (x)/*tan(x)|
2*|\1 + tan (x)/*\1 + 3*tan (x)/ - -------- + --------------- - ----------------------|
  |                                       3              2              2 + x         |
  \                                (2 + x)        (2 + x)                             /
---------------------------------------------------------------------------------------
                                         2 + x                                         
$$\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x + 2} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(x + 2\right)^{2}} - \frac{3 \tan{\left(x \right)}}{\left(x + 2\right)^{3}}\right)}{x + 2}$$
The graph
Derivative of tan(x)/(x+2)