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 −4-4 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:

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

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-100100
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