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

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

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

You have entered [src] 
   /        2\
   | 2*x - x |
cos\E        /
cos⁡(e−x2+2x)\cos{\left(e^{- x^{2} + 2 x} \right)} 
cos(E^(2*x - x^2))
Detail solution

  1. Let u=e−x2+2xu = e^{- x^{2} + 2 x}.

  2. The derivative of cosine is negative sine:

    dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

  3. Then, apply the chain rule. Multiply by ddxe−x2+2x\frac{d}{d x} e^{- x^{2} + 2 x}:

    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 of the chain rule is:

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

  4. Now simplify:

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

The answer is:

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

The graph
-1.0-0.8-0.6-0.4-0.21.00.00.20.40.60.85-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                   2    /        2\
            2*x - x     | 2*x - x |
-(2 - 2*x)*e        *sin\E        /
−(2−2x)e−x2+2xsin⁡(e−x2+2x)- \left(2 - 2 x\right) e^{- x^{2} + 2 x} \sin{\left(e^{- x^{2} + 2 x} \right)}
Simplify
The second derivative [src] 
  /            2    / x*(2 - x)\             2    / x*(2 - x)\  x*(2 - x)      / x*(2 - x)\\  x*(2 - x)
2*\- 2*(-1 + x) *sin\e         / - 2*(-1 + x) *cos\e         /*e          + sin\e         //*e
2(−2(x−1)2ex(2−x)cos⁡(ex(2−x))−2(x−1)2sin⁡(ex(2−x))+sin⁡(ex(2−x)))ex(2−x)2 \left(- 2 \left(x - 1\right)^{2} e^{x \left(2 - x\right)} \cos{\left(e^{x \left(2 - x\right)} \right)} - 2 \left(x - 1\right)^{2} \sin{\left(e^{x \left(2 - x\right)} \right)} + \sin{\left(e^{x \left(2 - x\right)} \right)}\right) e^{x \left(2 - x\right)}
Simplify
The third derivative [src] 
           /       / x*(2 - x)\        / x*(2 - x)\  x*(2 - x)             2    / x*(2 - x)\             2  2*x*(2 - x)    / x*(2 - x)\             2    / x*(2 - x)\  x*(2 - x)\  x*(2 - x)
4*(-1 + x)*\- 3*sin\e         / - 3*cos\e         /*e          + 2*(-1 + x) *sin\e         / - 2*(-1 + x) *e           *sin\e         / + 6*(-1 + x) *cos\e         /*e         /*e
4(x−1)(−2(x−1)2e2x(2−x)sin⁡(ex(2−x))+6(x−1)2ex(2−x)cos⁡(ex(2−x))+2(x−1)2sin⁡(ex(2−x))−3ex(2−x)cos⁡(ex(2−x))−3sin⁡(ex(2−x)))ex(2−x)4 \left(x - 1\right) \left(- 2 \left(x - 1\right)^{2} e^{2 x \left(2 - x\right)} \sin{\left(e^{x \left(2 - x\right)} \right)} + 6 \left(x - 1\right)^{2} e^{x \left(2 - x\right)} \cos{\left(e^{x \left(2 - x\right)} \right)} + 2 \left(x - 1\right)^{2} \sin{\left(e^{x \left(2 - x\right)} \right)} - 3 e^{x \left(2 - x\right)} \cos{\left(e^{x \left(2 - x\right)} \right)} - 3 \sin{\left(e^{x \left(2 - x\right)} \right)}\right) e^{x \left(2 - x\right)}
Simplify
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