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Integral of d{x}:
x^4/(x^2+1)
How do you in partial fractions?:
x^4/(x^2+1)
Identical expressions
x^ four /(x^ two + one)
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x to the power of four divide by (x to the power of two plus one)
x4/(x2+1)
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x^4/x^2+1
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Similar expressions
x^4/(x^2-1)

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

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

The solution

You have entered [src]

   4  
  x   
------
 2    
x  + 1

$$\frac{x^{4}}{x^{2} + 1}$$

x^4/(x^2 + 1)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{4}$ and $g{\left(x \right)} = x^{2} + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{2} + 1$ term by term:
  1. The derivative of the constant $1$ 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:

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

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

The answer is:

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

The graph

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/

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

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

$$\frac{2 x^{2} \left(\frac{x^{2} \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1} - \frac{8 x^{2}}{x^{2} + 1} + 6\right)}{x^{2} + 1}$$

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

$$\frac{24 x \left(- \frac{x^{4} \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right)}{\left(x^{2} + 1\right)^{2}} + \frac{x^{2} \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1} - \frac{3 x^{2}}{x^{2} + 1} + 1\right)}{x^{2} + 1}$$

Simplify

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

The graph:

Piecewise:

The solution