Calculus Examples

y = x e^{x}

$y = x e^{x}$

Step 1

Differentiate using the Product Rule which states that

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ is

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

f (x) = x

$f (x) = x$ and

g (x) = e^{x}

$g (x) = e^{x}$ .

x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x]

$x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x]$

Step 2

Differentiate using the Exponential Rule which states that

\frac{d}{d x} [a^{x}]

$\frac{d}{d x} [a^{x}]$ is

$a^{x} \ln (a)$ where

$a$ =

$e$ .

$x e^{x} + e^{x} \frac{d}{d x} [x]$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$x e^{x} + e^{x} \cdot 1$

Step 3.2

Multiply

$e^{x}$ by

$1$ .

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

$y = x e^{x}$

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