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The derivative of the function/ e^x(x+1)

Derivative of e^x(x+1)

The solution

You have entered [src]

 x        
e *(x + 1)

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

d / x        \
--\e *(x + 1)/
dx

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

Transform

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = e^{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $e^{x}$ is itself.
$g{\left(x \right)} = x + 1$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x + 1$ term by term:
  1. Apply the power rule: $x$ goes to $1$
  2. The derivative of the constant $1$ is zero.
  The result is: $1$
The result is: $\left(x + 1\right) e^{x} + e^{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 x            x
e  + (x + 1)*e

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

Simplify

The second derivative [src]

         x
(3 + x)*e

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

Simplify

The third derivative [src]

         x
(4 + x)*e

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

Simplify

The graph

Derivative of e^x(x+1)