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Derivative of x^3/(x^2+5)
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Integral of d{x}:
x^3/(x^2+5)
Graphing y =:
x^3/(x^2+5)
Identical expressions
x^ three /(x^ two + five)
x cubed divide by (x squared plus 5)
x to the power of three divide by (x to the power of two plus five)
x3/(x2+5)
x3/x2+5
x³/(x²+5)
x to the power of 3/(x to the power of 2+5)
x^3/x^2+5
x^3 divide by (x^2+5)
Similar expressions
x^3/(x^2-5)

The derivative of the function/ x^3/(x^2+5)

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

The solution

You have entered [src]

   3  
  x   
------
 2    
x  + 5

$$\frac{x^{3}}{x^{2} + 5}$$

x^3/(x^2 + 5)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x^{3}$ and $g{\left(x \right)} = x^{2} + 5$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{2} + 5$ term by term:
  1. The derivative of the constant $5$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x$
Now plug in to the quotient rule:

$\frac{- 2 x^{4} + 3 x^{2} \left(x^{2} + 5\right)}{\left(x^{2} + 5\right)^{2}}$
Now simplify:

$\frac{x^{2} \left(x^{2} + 15\right)}{\left(x^{2} + 5\right)^{2}}$

The answer is:

$\frac{x^{2} \left(x^{2} + 15\right)}{\left(x^{2} + 5\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        4         2 
     2*x       3*x  
- --------- + ------
          2    2    
  / 2    \    x  + 5
  \x  + 5/

$$- \frac{2 x^{4}}{\left(x^{2} + 5\right)^{2}} + \frac{3 x^{2}}{x^{2} + 5}$$

Simplify

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

$$\frac{2 x \left(\frac{x^{2} \left(\frac{4 x^{2}}{x^{2} + 5} - 1\right)}{x^{2} + 5} - \frac{6 x^{2}}{x^{2} + 5} + 3\right)}{x^{2} + 5}$$

Simplify

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

$$\frac{6 \left(- \frac{4 x^{4} \left(\frac{2 x^{2}}{x^{2} + 5} - 1\right)}{\left(x^{2} + 5\right)^{2}} + \frac{3 x^{2} \left(\frac{4 x^{2}}{x^{2} + 5} - 1\right)}{x^{2} + 5} - \frac{6 x^{2}}{x^{2} + 5} + 1\right)}{x^{2} + 5}$$

Simplify

The graph

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

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The solution