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

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

The solution

You have entered [src]

   /        2\
   | 2*x - x |
cos\E        /

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

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

Detail solution

Let $u = e^{- x^{2} + 2 x}$ .
The derivative of cosine is negative sine:

$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} e^{- x^{2} + 2 x}$ :
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 of the chain rule is:

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

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

The answer is:

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

The graph

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        /

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

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

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

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

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

Simplify

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

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The solution