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Identical expressions
exp^(x^ two +2x- one)
exponent of to the power of (x squared plus 2x minus 1)
exponent of to the power of (x to the power of two plus 2x minus one)
exp(x2+2x-1)
expx2+2x-1
exp^(x²+2x-1)
exp to the power of (x to the power of 2+2x-1)
exp^x^2+2x-1
Similar expressions
exp^(x^2+2x+1)
exp^(x^2-2x-1)

The derivative of the function/ exp^(x^2+2x-1)

Derivative of exp^(x^2+2x-1)

The solution

You have entered [src]

  2          
 x  + 2*x - 1
E

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

E^(x^2 + 2*x - 1)

Detail solution

Let $u = \left(x^{2} + 2 x\right) - 1$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x^{2} + 2 x\right) - 1\right)$ :
1. Differentiate $\left(x^{2} + 2 x\right) - 1$ term by term:
  1. Differentiate $x^{2} + 2 x$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    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: $2 x + 2$
  2. The derivative of the constant $-1$ is zero.
  The result is: $2 x + 2$
The result of the chain rule is:

$\left(2 x + 2\right) e^{\left(x^{2} + 2 x\right) - 1}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

5-я производная [src]

                                                 2      
          /              4             2\  -1 + x  + 2*x
8*(1 + x)*\15 + 4*(1 + x)  + 20*(1 + x) /*e

$$8 \left(x + 1\right) \left(4 \left(x + 1\right)^{4} + 20 \left(x + 1\right)^{2} + 15\right) e^{x^{2} + 2 x - 1}$$

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of exp^(x^2+2x-1)

The graph:

Piecewise:

The solution