Mister Exam

Other calculators


x^3/(x-2)^2

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

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
    3   
   x    
--------
       2
(x - 2) 
$$\frac{x^{3}}{\left(x - 2\right)^{2}}$$
x^3/(x - 2)^2
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Apply the power rule: goes to

    To find :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. Differentiate term by term:

        1. The derivative of the constant is zero.

        2. Apply the power rule: goes to

        The result is:

      The result of the chain rule is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
     2      3          
  3*x      x *(4 - 2*x)
-------- + ------------
       2            4  
(x - 2)      (x - 2)   
$$\frac{x^{3} \left(4 - 2 x\right)}{\left(x - 2\right)^{4}} + \frac{3 x^{2}}{\left(x - 2\right)^{2}}$$
The second derivative [src]
    /         2            \
    |        x        2*x  |
6*x*|1 + --------- - ------|
    |            2   -2 + x|
    \    (-2 + x)          /
----------------------------
                 2          
         (-2 + x)           
$$\frac{6 x \left(\frac{x^{2}}{\left(x - 2\right)^{2}} - \frac{2 x}{x - 2} + 1\right)}{\left(x - 2\right)^{2}}$$
The third derivative [src]
  /                   3           2  \
  |     6*x        4*x         9*x   |
6*|1 - ------ - --------- + ---------|
  |    -2 + x           3           2|
  \             (-2 + x)    (-2 + x) /
--------------------------------------
                      2               
              (-2 + x)                
$$\frac{6 \left(- \frac{4 x^{3}}{\left(x - 2\right)^{3}} + \frac{9 x^{2}}{\left(x - 2\right)^{2}} - \frac{6 x}{x - 2} + 1\right)}{\left(x - 2\right)^{2}}$$
The graph
Derivative of x^3/(x-2)^2