Calculus Examples

$\frac{d}{d x} (\frac{e^{x}}{\sqrt{x}})$

Step 1

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

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

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = e^{x}$ and $g (x) = x^{\frac{1}{2}}$ .

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

Step 3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Simplify.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

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

Step 10.2

Subtract $2$ from $1$ .

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

Step 11

Move the negative in front of the fraction.

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

Simplify.

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Step 15.1

Reorder terms.

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

Step 15.2

Simplify the numerator.

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Step 15.2.1

Factor $e^{x}$ out of $e^{x} x^{\frac{1}{2}} - \frac{e^{x}}{2 x^{\frac{1}{2}}}$ .

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Step 15.2.1.1

Factor $e^{x}$ out of $e^{x} x^{\frac{1}{2}}$ .

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

Step 15.2.1.2

Factor $e^{x}$ out of $- \frac{e^{x}}{2 x^{\frac{1}{2}}}$ .

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

Step 15.2.1.3

Factor $e^{x}$ out of $e^{x} (x^{\frac{1}{2}}) + e^{x} (- \frac{1}{2 x^{\frac{1}{2}}})$ .

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

Step 15.2.2

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

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

Step 15.2.3

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

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

Step 15.2.4

Combine the numerators over the common denominator.

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

Step 15.2.5

Simplify the numerator.

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Step 15.2.5.1

Rewrite using the commutative property of multiplication.

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

Step 15.2.5.2

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 15.2.5.2.1

Move $x^{\frac{1}{2}}$ .

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

Step 15.2.5.2.2

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

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

Step 15.2.5.2.3

Combine the numerators over the common denominator.

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

Step 15.2.5.2.4

Add $1$ and $1$ .

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

Step 15.2.5.2.5

Divide $2$ by $2$ .

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

Step 15.2.5.3

Simplify $2 x^{1}$ .

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

Step 15.3

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

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

Step 15.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 15.5

Combine.

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

Step 15.6

Multiply $x^{\frac{1}{2}}$ by $x$ by adding the exponents.

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Step 15.6.1

Move $x$ .

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

Step 15.6.2

Multiply $x$ by $x^{\frac{1}{2}}$ .

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Step 15.6.2.1

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

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

Step 15.6.2.2

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

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

Step 15.6.3

Write $1$ as a fraction with a common denominator.

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

Step 15.6.4

Combine the numerators over the common denominator.

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

Step 15.6.5

Add $2$ and $1$ .

$\frac{e^{x} (2 x - 1) \cdot 1}{2 x^{\frac{3}{2}}}$

Step 15.7

Multiply $e^{x} (2 x - 1)$ by $1$ .

$\frac{e^{x} (2 x - 1)}{2 x^{\frac{3}{2}}}$