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Similar expressions
2*exp/(2+x)^2

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /      4          6    \  x
2*|1 - ------ + ---------|*e 
  |    -2 + x           2|   
  \             (-2 + x) /   
-----------------------------
                  2          
          (-2 + x)

$$\frac{2 \left(1 - \frac{4}{x - 2} + \frac{6}{\left(x - 2\right)^{2}}\right) e^{x}}{\left(x - 2\right)^{2}}$$

Simplify

The third derivative [src]

  /        24        6          18   \  x
2*|1 - --------- - ------ + ---------|*e 
  |            3   -2 + x           2|   
  \    (-2 + x)             (-2 + x) /   
-----------------------------------------
                        2                
                (-2 + x)

$$\frac{2 \left(1 - \frac{6}{x - 2} + \frac{18}{\left(x - 2\right)^{2}} - \frac{24}{\left(x - 2\right)^{3}}\right) e^{x}}{\left(x - 2\right)^{2}}$$

Simplify

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

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The solution