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Derivative of:
Derivative of 6
Derivative of x*e^x^2
Derivative of (x+4)^2*(x+1)+9
Derivative of (x+3)/(x-1)
Graphing y =:
x*e^x^2
Integral of d{x}:
x*e^x^2
Identical expressions
x*e^x^ two
x multiply by e to the power of x squared
x multiply by e to the power of x to the power of two
x*ex2
x*e^x²
x*e to the power of x to the power of 2
xe^x^2
xex2

The derivative of the function/ x*e^x^2

Derivative of x*e^x^2

The solution

You have entered [src]

   / 2\
   \x /
x*E

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

x*E^(x^2)

Detail solution

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: $e^{x^{2}} + 2 x^{2} e^{x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 / 2\         / 2\
 \x /      2  \x /
E     + 2*x *e

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of x*e^x^2