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3*x/(x^2+1)
The derivative of the function/ 3*x/(x^2+1)

Derivative of 3*x/(x^2+1)

Function f() - derivative -N order at the point
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from to

Piecewise:

The solution

You have entered [src] 
 3*x  
------
 2    
x  + 1
3xx2+1\frac{3 x}{x^{2} + 1} 
(3*x)/(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)=3xf{\left(x \right)} = 3 x and g(x)=x2+1g{\left(x \right)} = x^{2} + 1.

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

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Apply the power rule: xx goes to 11

      So, the result is: 33

    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:

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

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
               2  
  3         6*x   
------ - ---------
 2               2
x  + 1   / 2    \ 
         \x  + 1/
−6x2(x2+1)2+3x2+1- \frac{6 x^{2}}{\left(x^{2} + 1\right)^{2}} + \frac{3}{x^{2} + 1}
Simplify
The second derivative [src] 
    /         2 \
    |      4*x  |
6*x*|-3 + ------|
    |          2|
    \     1 + x /
-----------------
            2    
    /     2\     
    \1 + x /
6x(4x2x2+1−3)(x2+1)2\frac{6 x \left(\frac{4 x^{2}}{x^{2} + 1} - 3\right)}{\left(x^{2} + 1\right)^{2}}
Simplify
The third derivative [src] 
   /                   /         2 \\
   |                 2 |      2*x  ||
   |              4*x *|-1 + ------||
   |         2         |          2||
   |      4*x          \     1 + x /|
18*|-1 + ------ - ------------------|
   |          2              2      |
   \     1 + x          1 + x       /
-------------------------------------
                      2              
              /     2\               
              \1 + x /
18(−4x2(2x2x2+1−1)x2+1+4x2x2+1−1)(x2+1)2\frac{18 \left(- \frac{4 x^{2} \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1} + \frac{4 x^{2}}{x^{2} + 1} - 1\right)}{\left(x^{2} + 1\right)^{2}}
Simplify
The graph
Derivative of 3*x/(x^2+1)
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