Calculus Examples

$y = x^{e} e^{x}$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{e}$ and $g (x) = e^{x}$ .

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

Step 2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = e$ .

$x^{e} e^{x} + e^{x} (e x^{e - 1})$

Step 4

Multiply $e^{x}$ by $e$ by adding the exponents.

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Step 4.1

Move $e$ .

$x^{e} e^{x} + e e^{x} x^{e - 1}$

Step 4.2

Multiply $e$ by $e^{x}$ .

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Step 4.2.1

Raise $e$ to the power of $1$ .

$x^{e} e^{x} + e^{1} e^{x} x^{e - 1}$

Step 4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{e} e^{x} + e^{1 + x} x^{e - 1}$

Step 5

Reorder terms.

$e^{x} x^{e} + e^{1 + x} x^{e - 1}$