Mister Exam

Other calculators

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

Derivative of x/(x^2+4)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  x   
------
 2    
x  + 4
xx2+4\frac{x}{x^{2} + 4} 
x/(x^2 + 4)
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)=xf{\left(x \right)} = x and g(x)=x2+4g{\left(x \right)} = x^{2} + 4.

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

    1. Apply the power rule: xx goes to 11

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

    1. Differentiate x2+4x^{2} + 4 term by term:

      1. The derivative of the constant 44 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−x2(x2+4)2\frac{4 - x^{2}}{\left(x^{2} + 4\right)^{2}}

The answer is:

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

The graph
02468-8-6-4-2-10100.5-0.5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
               2  
  1         2*x   
------ - ---------
 2               2
x  + 4   / 2    \ 
         \x  + 4/
−2x2(x2+4)2+1x2+4- \frac{2 x^{2}}{\left(x^{2} + 4\right)^{2}} + \frac{1}{x^{2} + 4}
Simplify
The second derivative [src] 
    /         2 \
    |      4*x  |
2*x*|-3 + ------|
    |          2|
    \     4 + x /
-----------------
            2    
    /     2\     
    \4 + x /
2x(4x2x2+4−3)(x2+4)2\frac{2 x \left(\frac{4 x^{2}}{x^{2} + 4} - 3\right)}{\left(x^{2} + 4\right)^{2}}
Simplify
The third derivative [src] 
  /                   /         2 \\
  |                 2 |      2*x  ||
  |              4*x *|-1 + ------||
  |         2         |          2||
  |      4*x          \     4 + x /|
6*|-1 + ------ - ------------------|
  |          2              2      |
  \     4 + x          4 + x       /
------------------------------------
                     2              
             /     2\               
             \4 + x /
6(−4x2(2x2x2+4−1)x2+4+4x2x2+4−1)(x2+4)2\frac{6 \left(- \frac{4 x^{2} \left(\frac{2 x^{2}}{x^{2} + 4} - 1\right)}{x^{2} + 4} + \frac{4 x^{2}}{x^{2} + 4} - 1\right)}{\left(x^{2} + 4\right)^{2}}
Simplify
The graph
Derivative of x/(x^2+4)
    © Mister Exam – Calculator