Mister Exam

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The derivative of the function/ x*e^x

Derivative of x*e^x

The solution

You have entered [src]

   x
x*E

$$e^{x} x$$

x*E^x

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 $1$
$g{\left(x \right)} = e^{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $e^{x}$ is itself.
The result is: $e^{x} + x e^{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 x      x
E  + x*e

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

Simplify

The second derivative [src]

         x
(2 + x)*e

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

Simplify

The third derivative [src]

         x
(3 + x)*e

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

Simplify

The graph

Derivative of x*e^x