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The derivative of the function/ x^4/(x^2+1)

Derivative of x^4/(x^2+1)

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

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The solution

You have entered [src] 
   4  
  x   
------
 2    
x  + 1
x4x2+1\frac{x^{4}}{x^{2} + 1} 
x^4/(x^2 + 1)
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)=x4f{\left(x \right)} = x^{4} and g(x)=x2+1g{\left(x \right)} = x^{2} + 1.

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

    1. Apply the power rule: x4x^{4} goes to 4x34 x^{3}

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

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

      1. The derivative of the constant 11 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:

    −2x5+4x3(x2+1)(x2+1)2\frac{- 2 x^{5} + 4 x^{3} \left(x^{2} + 1\right)}{\left(x^{2} + 1\right)^{2}}

  2. Now simplify:

    2x3(x2+2)(x2+1)2\frac{2 x^{3} \left(x^{2} + 2\right)}{\left(x^{2} + 1\right)^{2}}

The answer is:

2x3(x2+2)(x2+1)2\frac{2 x^{3} \left(x^{2} + 2\right)}{\left(x^{2} + 1\right)^{2}}

The graph
1.09.02.03.04.05.06.07.08.00100
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        5         3 
     2*x       4*x  
- --------- + ------
          2    2    
  / 2    \    x  + 1
  \x  + 1/
−2x5(x2+1)2+4x3x2+1- \frac{2 x^{5}}{\left(x^{2} + 1\right)^{2}} + \frac{4 x^{3}}{x^{2} + 1}
Simplify
The second derivative [src] 
     /                /         2 \\
     |              2 |      4*x  ||
     |             x *|-1 + ------||
     |        2       |          2||
   2 |     8*x        \     1 + x /|
2*x *|6 - ------ + ----------------|
     |         2             2     |
     \    1 + x         1 + x      /
------------------------------------
                    2               
               1 + x
2x2(x2(4x2x2+1−1)x2+1−8x2x2+1+6)x2+1\frac{2 x^{2} \left(\frac{x^{2} \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1} - \frac{8 x^{2}}{x^{2} + 1} + 6\right)}{x^{2} + 1}
Simplify
The third derivative [src] 
     /                /         2 \      /         2 \\
     |              2 |      4*x  |    4 |      2*x  ||
     |             x *|-1 + ------|   x *|-1 + ------||
     |        2       |          2|      |          2||
     |     3*x        \     1 + x /      \     1 + x /|
24*x*|1 - ------ + ---------------- - ----------------|
     |         2             2                   2    |
     |    1 + x         1 + x            /     2\     |
     \                                   \1 + x /     /
-------------------------------------------------------
                              2                        
                         1 + x
24x(−x4(2x2x2+1−1)(x2+1)2+x2(4x2x2+1−1)x2+1−3x2x2+1+1)x2+1\frac{24 x \left(- \frac{x^{4} \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right)}{\left(x^{2} + 1\right)^{2}} + \frac{x^{2} \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1} - \frac{3 x^{2}}{x^{2} + 1} + 1\right)}{x^{2} + 1}
Simplify
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