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

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

Function f() - derivative -N order at the point
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The graph:

from to

Piecewise:

The solution

You have entered [src] 
   2  
  x   
------
 2    
x  + 3
x2x2+3\frac{x^{2}}{x^{2} + 3} 
x^2/(x^2 + 3)
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)=x2f{\left(x \right)} = x^{2} and g(x)=x2+3g{\left(x \right)} = x^{2} + 3.

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

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

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

    1. Differentiate x2+3x^{2} + 3 term by term:

      1. The derivative of the constant 33 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:

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

  2. Now simplify:

    6x(x2+3)2\frac{6 x}{\left(x^{2} + 3\right)^{2}}

The answer is:

6x(x2+3)2\frac{6 x}{\left(x^{2} + 3\right)^{2}}

The graph
02468-8-6-4-2-10101-1
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        3           
     2*x       2*x  
- --------- + ------
          2    2    
  / 2    \    x  + 3
  \x  + 3/
−2x3(x2+3)2+2xx2+3- \frac{2 x^{3}}{\left(x^{2} + 3\right)^{2}} + \frac{2 x}{x^{2} + 3}
Simplify
The second derivative [src] 
  /                /         2 \\
  |              2 |      4*x  ||
  |             x *|-1 + ------||
  |        2       |          2||
  |     4*x        \     3 + x /|
2*|1 - ------ + ----------------|
  |         2             2     |
  \    3 + x         3 + x      /
---------------------------------
                   2             
              3 + x
2(x2(4x2x2+3−1)x2+3−4x2x2+3+1)x2+3\frac{2 \left(\frac{x^{2} \left(\frac{4 x^{2}}{x^{2} + 3} - 1\right)}{x^{2} + 3} - \frac{4 x^{2}}{x^{2} + 3} + 1\right)}{x^{2} + 3}
Simplify
The third derivative [src] 
     /                   /         2 \\
     |                 2 |      2*x  ||
     |              2*x *|-1 + ------||
     |         2         |          2||
     |      4*x          \     3 + x /|
12*x*|-2 + ------ - ------------------|
     |          2              2      |
     \     3 + x          3 + x       /
---------------------------------------
                       2               
               /     2\                
               \3 + x /
12x(−2x2(2x2x2+3−1)x2+3+4x2x2+3−2)(x2+3)2\frac{12 x \left(- \frac{2 x^{2} \left(\frac{2 x^{2}}{x^{2} + 3} - 1\right)}{x^{2} + 3} + \frac{4 x^{2}}{x^{2} + 3} - 2\right)}{\left(x^{2} + 3\right)^{2}}
Simplify
The graph
Derivative of x^2/(x^2+3)
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