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Derivative of:
Derivative of x^-6
Derivative of (x+3)/(x-3)
Derivative of (x^3)'
Derivative of log(x^4+x)
Identical expressions
exp^(two *x-x^ two)
exponent of to the power of (2 multiply by x minus x squared )
exponent of to the power of (two multiply by x minus x to the power of two)
exp(2*x-x2)
exp2*x-x2
exp^(2*x-x²)
exp to the power of (2*x-x to the power of 2)
exp^(2x-x^2)
exp(2x-x2)
exp2x-x2
exp^2x-x^2
Similar expressions
exp^(2*x+x^2)

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

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

The solution

You have entered [src]

        2
 2*x - x 
E

$$e^{- x^{2} + 2 x}$$

E^(2*x - x^2)

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify