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(two - two x)exp^(2x-x^2)
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(two minus two x) exponent of to the power of (2x minus x squared )
(2-2x)exp(2x-x2)
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(2-2x)exp to the power of (2x-x to the power of 2)
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(2+2x)exp^(2x-x^2)
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The derivative of the function/ (2-2x)exp^(2x-x^2)

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

The solution

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

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

(2 - 2*x)*E^(2*x - x^2)

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /              2             2 /               2\\  -x*(-2 + x)
4*\3 - 6*(-1 + x)  + 2*(-1 + x) *\-3 + 2*(-1 + x) //*e

$$4 \left(2 \left(x - 1\right)^{2} \left(2 \left(x - 1\right)^{2} - 3\right) - 6 \left(x - 1\right)^{2} + 3\right) e^{- x \left(x - 2\right)}$$

Simplify

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

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The solution