Mister Exam

Lang:

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

Derivative of xexp(x^2)

The solution

You have entered [src]

   / 2\
   \x /
x*e

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

x*exp(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: $2 x^{2} e^{x^{2}} + e^{x^{2}}$
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\
   2  \x /    \x /
2*x *e     + e

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

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