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Derivative of:
Derivative of shx
Derivative of 2xe^(x^2)
Derivative of g(x)=8-4x^3+2x^8
Derivative of 3sin^2x-sin^3x
Integral of d{x}:
2xe^(x^2)
2xe^(x^2)
Identical expressions
two xe^(x^2)
2xe to the power of (x squared )
two xe to the power of (x squared )
2xe(x2)
2xex2
2xe^(x²)
2xe to the power of (x to the power of 2)
2xe^x^2

The derivative of the function/ 2xe^(x^2)

Derivative of 2xe^(x^2)

The solution

You have entered [src]

     / 2\
     \x /
2*x*e

$$2 x e^{x^{2}}$$

  /     / 2\\
d |     \x /|
--\2*x*e    /
dx

$$\frac{d}{d x} 2 x e^{x^{2}}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = e^{x^{2}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = x^{2}$ .
  2. The derivative of $e^{u}$ is itself.
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{2}$ :
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    The result of the chain rule is:
    
    $2 x e^{x^{2}}$
  The result is: $2 x^{2} e^{x^{2}} + e^{x^{2}}$
So, the result is: $4 x^{2} e^{x^{2}} + 2 e^{x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   / 2\         / 2\
   \x /      2  \x /
2*e     + 4*x *e

$$4 x^{2} e^{x^{2}} + 2 e^{x^{2}}$$

Simplify

The second derivative [src]

                / 2\
    /       2\  \x /
4*x*\3 + 2*x /*e

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

Simplify

The third derivative [src]

                                / 2\
  /       2      2 /       2\\  \x /
4*\3 + 6*x  + 2*x *\3 + 2*x //*e

$$4 \cdot \left(2 x^{2} \cdot \left(2 x^{2} + 3\right) + 6 x^{2} + 3\right) e^{x^{2}}$$

Simplify

The graph

Derivative of 2xe^(x^2)