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tan(x)/x^3

Derivative of tan(x)/x^3

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
tan(x)
------
   3  
  x   
$$\frac{\tan{\left(x \right)}}{x^{3}}$$
tan(x)/x^3
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. Apply the power rule: goes to

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
       2              
1 + tan (x)   3*tan(x)
----------- - --------
      3           4   
     x           x    
$$\frac{\tan^{2}{\left(x \right)} + 1}{x^{3}} - \frac{3 \tan{\left(x \right)}}{x^{4}}$$
The second derivative [src]
  /                         /       2   \           \
  |/       2   \          3*\1 + tan (x)/   6*tan(x)|
2*|\1 + tan (x)/*tan(x) - --------------- + --------|
  |                              x              2   |
  \                                            x    /
-----------------------------------------------------
                           3                         
                          x                          
$$\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{6 \tan{\left(x \right)}}{x^{2}}\right)}{x^{3}}$$
The third derivative [src]
  /                                               /       2   \     /       2   \       \
  |/       2   \ /         2   \   30*tan(x)   18*\1 + tan (x)/   9*\1 + tan (x)/*tan(x)|
2*|\1 + tan (x)/*\1 + 3*tan (x)/ - --------- + ---------------- - ----------------------|
  |                                     3              2                    x           |
  \                                    x              x                                 /
-----------------------------------------------------------------------------------------
                                             3                                           
                                            x                                            
$$\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) - \frac{9 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} + \frac{18 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2}} - \frac{30 \tan{\left(x \right)}}{x^{3}}\right)}{x^{3}}$$
The graph
Derivative of tan(x)/x^3