Calculus Examples

$f (x) = \sqrt{x} e^{x}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{d}{d x} [x^{\frac{1}{2}} e^{x}]$

Step 2

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 $f (x) = x^{\frac{1}{2}}$ and $g (x) = e^{x}$ .

$x^{\frac{1}{2}} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 3

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

$x^{\frac{1}{2}} e^{x} + e^{x} \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$x^{\frac{1}{2}} e^{x} + e^{x} (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{\frac{1}{2}} e^{x} + e^{x} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 6

Combine $- 1$ and $\frac{2}{2}$ .

$x^{\frac{1}{2}} e^{x} + e^{x} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 7

Combine the numerators over the common denominator.

$x^{\frac{1}{2}} e^{x} + e^{x} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$x^{\frac{1}{2}} e^{x} + e^{x} (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 8.2

Subtract $2$ from $1$ .

$x^{\frac{1}{2}} e^{x} + e^{x} (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 9

Move the negative in front of the fraction.

$x^{\frac{1}{2}} e^{x} + e^{x} (\frac{1}{2} x^{- \frac{1}{2}})$

Step 10

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$x^{\frac{1}{2}} e^{x} + e^{x} \frac{x^{- \frac{1}{2}}}{2}$

Step 11

Combine $e^{x}$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$x^{\frac{1}{2}} e^{x} + \frac{e^{x} x^{- \frac{1}{2}}}{2}$

Step 12

Simplify the expression.

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Step 12.1

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x^{\frac{1}{2}} e^{x} + \frac{e^{x}}{2 x^{\frac{1}{2}}}$

Step 12.2

Reorder terms.

$e^{x} x^{\frac{1}{2}} + \frac{e^{x}}{2 x^{\frac{1}{2}}}$