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

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

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

The graph:

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Piecewise:

The solution

You have entered [src] 
   3  
  x   
------
 2    
x  + 5
x3x2+5\frac{x^{3}}{x^{2} + 5} 
x^3/(x^2 + 5)
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=x3f{\left(x \right)} = x^{3} and g(x)=x2+5g{\left(x \right)} = x^{2} + 5.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: x3x^{3} goes to 3x23 x^{2}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate x2+5x^{2} + 5 term by term:

      1. The derivative of the constant 55 is zero.

      2. Apply the power rule: x2x^{2} goes to 2x2 x

      The result is: 2x2 x

    Now plug in to the quotient rule:

    −2x4+3x2(x2+5)(x2+5)2\frac{- 2 x^{4} + 3 x^{2} \left(x^{2} + 5\right)}{\left(x^{2} + 5\right)^{2}}

  2. Now simplify:

    x2(x2+15)(x2+5)2\frac{x^{2} \left(x^{2} + 15\right)}{\left(x^{2} + 5\right)^{2}}

The answer is:

x2(x2+15)(x2+5)2\frac{x^{2} \left(x^{2} + 15\right)}{\left(x^{2} + 5\right)^{2}}

The graph
02468-8-6-4-2-1010-2020
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/
−2x4(x2+5)2+3x2x2+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
2x(x2(4x2x2+5−1)x2+5−6x2x2+5+3)x2+5\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
6(−4x4(2x2x2+5−1)(x2+5)2+3x2(4x2x2+5−1)x2+5−6x2x2+5+1)x2+5\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
Derivative of x^3/(x^2+5)
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