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x^3/(x^2+5)

Derivative of x^3/(x^2+5)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   3  
  x   
------
 2    
x  + 5
$$\frac{x^{3}}{x^{2} + 5}$$
x^3/(x^2 + 5)
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Apply the power rule: goes to

    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]
        4         2 
     2*x       3*x  
- --------- + ------
          2    2    
  / 2    \    x  + 5
  \x  + 5/          
$$- \frac{2 x^{4}}{\left(x^{2} + 5\right)^{2}} + \frac{3 x^{2}}{x^{2} + 5}$$
The second derivative [src]
    /                /         2 \\
    |              2 |      4*x  ||
    |             x *|-1 + ------||
    |        2       |          2||
    |     6*x        \     5 + x /|
2*x*|3 - ------ + ----------------|
    |         2             2     |
    \    5 + x         5 + x      /
-----------------------------------
                    2              
               5 + x               
$$\frac{2 x \left(\frac{x^{2} \left(\frac{4 x^{2}}{x^{2} + 5} - 1\right)}{x^{2} + 5} - \frac{6 x^{2}}{x^{2} + 5} + 3\right)}{x^{2} + 5}$$
The third derivative [src]
  /                  /         2 \        /         2 \\
  |                4 |      2*x  |      2 |      4*x  ||
  |             4*x *|-1 + ------|   3*x *|-1 + ------||
  |        2         |          2|        |          2||
  |     6*x          \     5 + x /        \     5 + x /|
6*|1 - ------ - ------------------ + ------------------|
  |         2               2                   2      |
  |    5 + x        /     2\               5 + x       |
  \                 \5 + x /                           /
--------------------------------------------------------
                              2                         
                         5 + x                          
$$\frac{6 \left(- \frac{4 x^{4} \left(\frac{2 x^{2}}{x^{2} + 5} - 1\right)}{\left(x^{2} + 5\right)^{2}} + \frac{3 x^{2} \left(\frac{4 x^{2}}{x^{2} + 5} - 1\right)}{x^{2} + 5} - \frac{6 x^{2}}{x^{2} + 5} + 1\right)}{x^{2} + 5}$$
The graph
Derivative of x^3/(x^2+5)