Mister Exam

Other calculators

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

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

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
 4*x  
------
     2
1 + x
4xx2+1\frac{4 x}{x^{2} + 1} 
(4*x)/(1 + x^2)
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)=4xf{\left(x \right)} = 4 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: 44

    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:

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

  2. Now simplify:

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

The answer is:

4(1−x2)(x2+1)2\frac{4 \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  
  4         8*x   
------ - ---------
     2           2
1 + x    /     2\ 
         \1 + x /
−8x2(x2+1)2+4x2+1- \frac{8 x^{2}}{\left(x^{2} + 1\right)^{2}} + \frac{4}{x^{2} + 1}
Simplify
The second derivative [src] 
    /         2 \
    |      4*x  |
8*x*|-3 + ------|
    |          2|
    \     1 + x /
-----------------
            2    
    /     2\     
    \1 + x /
8x(4x2x2+1−3)(x2+1)2\frac{8 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 /|
24*|-1 + ------ - ------------------|
   |          2              2      |
   \     1 + x          1 + x       /
-------------------------------------
                      2              
              /     2\               
              \1 + x /
24(−4x2(2x2x2+1−1)x2+1+4x2x2+1−1)(x2+1)2\frac{24 \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
    © Mister Exam – Calculator