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

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

Function f() - derivative -N order at the point
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The solution

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                  2
           2*x - x 
(2 - 2*x)*E
e−x2+2x(2−2x)e^{- x^{2} + 2 x} \left(2 - 2 x\right) 
(2 - 2*x)*E^(2*x - x^2)
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=2−2xf{\left(x \right)} = 2 - 2 x; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Differentiate 2−2x2 - 2 x term by term:

      1. The derivative of the constant 22 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: xx goes to 11

        So, the result is: −2-2

      The result is: −2-2

    g(x)=e−x2+2xg{\left(x \right)} = e^{- x^{2} + 2 x}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=−x2+2xu = - x^{2} + 2 x.

    2. The derivative of eue^{u} is itself.

    3. Then, apply the chain rule. Multiply by ddx(−x2+2x)\frac{d}{d x} \left(- x^{2} + 2 x\right):

      1. Differentiate −x2+2x- 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: xx goes to 11

          So, the result is: 22

        2. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: x2x^{2} goes to 2x2 x

          So, the result is: −2x- 2 x

        The result is: 2−2x2 - 2 x

      The result of the chain rule is:

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

    The result is: (2−2x)2e−x2+2x−2e−x2+2x\left(2 - 2 x\right)^{2} e^{- x^{2} + 2 x} - 2 e^{- x^{2} + 2 x}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-1010
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The first derivative [src] 
            2                      2
     2*x - x             2  2*x - x 
- 2*e         + (2 - 2*x) *e
(2−2x)2e−x2+2x−2e−x2+2x\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(x−1)(−(2(x−1)2−1)e−x(x−2)+2ex(2−x))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(2(x−1)2(2(x−1)2−3)−6(x−1)2+3)e−x(x−2)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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