Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of tanx
Derivative of 2*x/(x-2)
Derivative of 10/(x+2)
Derivative of y=5sinx
Identical expressions
(two *x)/(x- two)^ two
(2 multiply by x) divide by (x minus 2) squared
(two multiply by x) divide by (x minus two) to the power of two
(2*x)/(x-2)2
2*x/x-22
(2*x)/(x-2)²
(2*x)/(x-2) to the power of 2
(2x)/(x-2)^2
(2x)/(x-2)2
2x/x-22
2x/x-2^2
(2*x) divide by (x-2)^2
Similar expressions
(2*x)/(x+2)^2

The derivative of the function/ (2*x)/(x-2)^2

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

The solution

You have entered [src]

  2*x   
--------
       2
(x - 2)

$$\frac{2 x}{\left(x - 2\right)^{2}}$$

(2*x)/(x - 2)^2

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 2 x$ and $g{\left(x \right)} = \left(x - 2\right)^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x$ goes to $1$
  So, the result is: $2$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x - 2$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 2\right)$ :
  1. Differentiate $x - 2$ term by term:
    1. The derivative of the constant $-2$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result of the chain rule is:
  
  $2 x - 4$
Now plug in to the quotient rule:

$\frac{- 2 x \left(2 x - 4\right) + 2 \left(x - 2\right)^{2}}{\left(x - 2\right)^{4}}$
Now simplify:

$- \frac{2 x + 4}{\left(x - 2\right)^{3}}$

The answer is:

$- \frac{2 x + 4}{\left(x - 2\right)^{3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2       2*x*(4 - 2*x)
-------- + -------------
       2             4  
(x - 2)       (x - 2)

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

Simplify

The second derivative [src]

  /      3*x  \
4*|-2 + ------|
  \     -2 + x/
---------------
           3   
   (-2 + x)

$$\frac{4 \left(\frac{3 x}{x - 2} - 2\right)}{\left(x - 2\right)^{3}}$$

Simplify

The third derivative [src]

   /     4*x  \
12*|3 - ------|
   \    -2 + x/
---------------
           4   
   (-2 + x)

$$\frac{12 \left(- \frac{4 x}{x - 2} + 3\right)}{\left(x - 2\right)^{4}}$$

Simplify