Calculus Examples

$f (x) = \ln (x e^{\sqrt{x}} + 5)$

Step 1

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

$\frac{d}{d x} [\ln (x e^{x^{\frac{1}{2}}} + 5)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x e^{x^{\frac{1}{2}}} + 5$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $x e^{x^{\frac{1}{2}}} + 5$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [x e^{x^{\frac{1}{2}}} + 5]$

Step 2.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$\frac{1}{u_{1}} \frac{d}{d x} [x e^{x^{\frac{1}{2}}} + 5]$

Step 2.3

Replace all occurrences of $u_{1}$ with $x e^{x^{\frac{1}{2}}} + 5$ .

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} \frac{d}{d x} [x e^{x^{\frac{1}{2}}} + 5]$

Step 3

By the Sum Rule, the derivative of $x e^{x^{\frac{1}{2}}} + 5$ with respect to $x$ is $\frac{d}{d x} [x e^{x^{\frac{1}{2}}}] + \frac{d}{d x} [5]$ .

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (\frac{d}{d x} [x e^{x^{\frac{1}{2}}}] + \frac{d}{d x} [5])$

Step 4

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

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (x \frac{d}{d x} [e^{x^{\frac{1}{2}}}] + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 5

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

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $x^{\frac{1}{2}}$ .

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (x (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [x^{\frac{1}{2}}]) + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 5.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (x (e^{u_{2}} \frac{d}{d x} [x^{\frac{1}{2}}]) + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 5.3

Replace all occurrences of $u_{2}$ with $x^{\frac{1}{2}}$ .

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (x (e^{x^{\frac{1}{2}}} \frac{d}{d x} [x^{\frac{1}{2}}]) + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 6

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

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (x (e^{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{1}{2} - 1})) + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 7

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

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (x (e^{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})) + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 8

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

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (x (e^{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})) + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 9

Combine the numerators over the common denominator.

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (x (e^{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})) + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (x (e^{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{1 - 2}{2}})) + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 10.2

Subtract $2$ from $1$ .

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (x (e^{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{- 1}{2}})) + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 11

Move the negative in front of the fraction.

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (x (e^{x^{\frac{1}{2}}} (\frac{1}{2} x^{- \frac{1}{2}})) + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 12

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

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (x (e^{x^{\frac{1}{2}}} \frac{x^{- \frac{1}{2}}}{2}) + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 13

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

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (x \frac{e^{x^{\frac{1}{2}}} x^{- \frac{1}{2}}}{2} + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 14

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

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (x \frac{e^{x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}}} + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 15

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

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (\frac{x e^{x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}}} + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 16

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

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (\frac{x e^{x^{\frac{1}{2}}} x^{- \frac{1}{2}}}{2} + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 17

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

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Step 17.1

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

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (\frac{x^{- \frac{1}{2}} x e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 17.2

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

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Step 17.2.1

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

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (\frac{x^{- \frac{1}{2}} x^{1} e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 17.2.2

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

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (\frac{x^{- \frac{1}{2} + 1} e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 17.3

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

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (\frac{x^{- \frac{1}{2} + \frac{2}{2}} e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 17.4

Combine the numerators over the common denominator.

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (\frac{x^{\frac{- 1 + 2}{2}} e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 17.5

Add $- 1$ and $2$ .

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (\frac{x^{\frac{1}{2}} e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 18

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

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (\frac{x^{\frac{1}{2}} e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}} \cdot 1 + \frac{d}{d x} [5])$

Step 19

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

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (\frac{x^{\frac{1}{2}} e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}} + \frac{d}{d x} [5])$

Step 20

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (\frac{x^{\frac{1}{2}} e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}} + 0)$

Step 21

Add $\frac{x^{\frac{1}{2}} e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}}$ and $0$ .

$\frac{1}{x e^{x^{\frac{1}{2}}} + 5} (\frac{x^{\frac{1}{2}} e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}})$

Step 22

Simplify.

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Step 22.1

Multiply $\frac{1}{x e^{x^{\frac{1}{2}}} + 5}$ by $\frac{x^{\frac{1}{2}} e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}}$ .

$\frac{\frac{x^{\frac{1}{2}} e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}}}{x e^{x^{\frac{1}{2}}} + 5}$

Step 22.2

Multiply the numerator and denominator of the fraction by $2$ .

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Step 22.2.1

Multiply $\frac{\frac{x^{\frac{1}{2}} e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}}}{x e^{x^{\frac{1}{2}}} + 5}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{\frac{x^{\frac{1}{2}} e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}}}{x e^{x^{\frac{1}{2}}} + 5}$

Step 22.2.2

Combine.

$\frac{2 (\frac{x^{\frac{1}{2}} e^{x^{\frac{1}{2}}}}{2} + e^{x^{\frac{1}{2}}})}{2 (x e^{x^{\frac{1}{2}}} + 5)}$

Step 22.3

Apply the distributive property.

$\frac{2 \frac{x^{\frac{1}{2}} e^{x^{\frac{1}{2}}}}{2} + 2 e^{x^{\frac{1}{2}}}}{2 (x e^{x^{\frac{1}{2}}}) + 2 \cdot 5}$

Step 22.4

Cancel the common factor of $2$ .

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Step 22.4.1

Cancel the common factor.

$\frac{2 \frac{x^{\frac{1}{2}} e^{x^{\frac{1}{2}}}}{2} + 2 e^{x^{\frac{1}{2}}}}{2 (x e^{x^{\frac{1}{2}}}) + 2 \cdot 5}$

Step 22.4.2

Rewrite the expression.

$\frac{x^{\frac{1}{2}} e^{x^{\frac{1}{2}}} + 2 e^{x^{\frac{1}{2}}}}{2 (x e^{x^{\frac{1}{2}}}) + 2 \cdot 5}$

Step 22.5

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

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Step 22.5.1

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

$\frac{e^{x^{\frac{1}{2}}} x^{\frac{1}{2}} + 2 e^{x^{\frac{1}{2}}}}{2 x e^{x^{\frac{1}{2}}} + 2 \cdot 5}$

Step 22.5.2

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

$\frac{e^{x^{\frac{1}{2}}} x^{\frac{1}{2}} + e^{x^{\frac{1}{2}}} \cdot 2}{2 x e^{x^{\frac{1}{2}}} + 2 \cdot 5}$

Step 22.5.3

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

$\frac{e^{x^{\frac{1}{2}}} (x^{\frac{1}{2}} + 2)}{2 x e^{x^{\frac{1}{2}}} + 2 \cdot 5}$

Step 22.6

Factor $2$ out of $2 x e^{x^{\frac{1}{2}}} + 2 \cdot 5$ .

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Step 22.6.1

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

$\frac{e^{x^{\frac{1}{2}}} (x^{\frac{1}{2}} + 2)}{2 (x e^{x^{\frac{1}{2}}}) + 2 \cdot 5}$

Step 22.6.2

Factor $2$ out of $2 \cdot 5$ .

$\frac{e^{x^{\frac{1}{2}}} (x^{\frac{1}{2}} + 2)}{2 (x e^{x^{\frac{1}{2}}}) + 2 (5)}$

Step 22.6.3

Factor $2$ out of $2 (x e^{x^{\frac{1}{2}}}) + 2 (5)$ .

$\frac{e^{x^{\frac{1}{2}}} (x^{\frac{1}{2}} + 2)}{2 (x e^{x^{\frac{1}{2}}} + 5)}$