Calculus Examples

$\frac{d}{d x} {(1 + \frac{8}{x})}^{x}$

Step 1

Use the properties of logarithms to simplify the differentiation.

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Step 1.1

Rewrite ${(1 + \frac{8}{x})}^{x}$ as $e^{\ln ({(1 + \frac{8}{x})}^{x})}$ .

$\frac{d}{d x} [e^{\ln ({(1 + \frac{8}{x})}^{x})}]$

Step 1.2

Expand $\ln ({(1 + \frac{8}{x})}^{x})$ by moving $x$ outside the logarithm.

$\frac{d}{d x} [e^{x \ln (1 + \frac{8}{x})}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x \ln (1 + \frac{8}{x})$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $x \ln (1 + \frac{8}{x})$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [x \ln (1 + \frac{8}{x})]$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [x \ln (1 + \frac{8}{x})]$

Step 2.3

Replace all occurrences of $u_{1}$ with $x \ln (1 + \frac{8}{x})$ .

$e^{x \ln (1 + \frac{8}{x})} \frac{d}{d x} [x \ln (1 + \frac{8}{x})]$

Step 3

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 $g (x) = \ln (1 + \frac{8}{x})$ .

$e^{x \ln (1 + \frac{8}{x})} (x \frac{d}{d x} [\ln (1 + \frac{8}{x})] + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 4

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 $g (x) = 1 + \frac{8}{x}$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $1 + \frac{8}{x}$ .

$e^{x \ln (1 + \frac{8}{x})} (x (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [1 + \frac{8}{x}]) + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 4.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$e^{x \ln (1 + \frac{8}{x})} (x (\frac{1}{u_{2}} \frac{d}{d x} [1 + \frac{8}{x}]) + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 4.3

Replace all occurrences of $u_{2}$ with $1 + \frac{8}{x}$ .

$e^{x \ln (1 + \frac{8}{x})} (x (\frac{1}{1 + \frac{8}{x}} \frac{d}{d x} [1 + \frac{8}{x}]) + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 5

Differentiate.

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Step 5.1

Combine $\frac{1}{1 + \frac{8}{x}}$ and $x$ .

$e^{x \ln (1 + \frac{8}{x})} (\frac{x}{1 + \frac{8}{x}} \frac{d}{d x} [1 + \frac{8}{x}] + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 5.2

By the Sum Rule, the derivative of $1 + \frac{8}{x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\frac{8}{x}]$ .

$e^{x \ln (1 + \frac{8}{x})} (\frac{x}{1 + \frac{8}{x}} (\frac{d}{d x} [1] + \frac{d}{d x} [\frac{8}{x}]) + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 5.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$e^{x \ln (1 + \frac{8}{x})} (\frac{x}{1 + \frac{8}{x}} (0 + \frac{d}{d x} [\frac{8}{x}]) + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 5.4

Add $0$ and $\frac{d}{d x} [\frac{8}{x}]$ .

$e^{x \ln (1 + \frac{8}{x})} (\frac{x}{1 + \frac{8}{x}} \frac{d}{d x} [\frac{8}{x}] + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 5.5

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8}{x}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{1}{x}]$ .

$e^{x \ln (1 + \frac{8}{x})} (\frac{x}{1 + \frac{8}{x}} (8 \frac{d}{d x} [\frac{1}{x}]) + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 5.6

Combine fractions.

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Step 5.6.1

Combine $8$ and $\frac{x}{1 + \frac{8}{x}}$ .

$e^{x \ln (1 + \frac{8}{x})} (\frac{8 x}{1 + \frac{8}{x}} \frac{d}{d x} [\frac{1}{x}] + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 5.6.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$e^{x \ln (1 + \frac{8}{x})} (\frac{8 x}{1 + \frac{8}{x}} \frac{d}{d x} [x^{- 1}] + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 5.7

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

$e^{x \ln (1 + \frac{8}{x})} (\frac{8 x}{1 + \frac{8}{x}} (- x^{- 2}) + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 5.8

Combine $\frac{8 x}{1 + \frac{8}{x}}$ and $x^{- 2}$ .

$e^{x \ln (1 + \frac{8}{x})} (- \frac{8 x \cdot x^{- 2}}{1 + \frac{8}{x}} + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 6

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

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Step 6.1

Move $x^{- 2}$ .

$e^{x \ln (1 + \frac{8}{x})} (- \frac{8 (x^{- 2} x)}{1 + \frac{8}{x}} + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 6.2

Multiply $x^{- 2}$ by $x$ .

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Step 6.2.1

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

$e^{x \ln (1 + \frac{8}{x})} (- \frac{8 (x^{- 2} x^{1})}{1 + \frac{8}{x}} + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 6.2.2

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

$e^{x \ln (1 + \frac{8}{x})} (- \frac{8 x^{- 2 + 1}}{1 + \frac{8}{x}} + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 6.3

Add $- 2$ and $1$ .

$e^{x \ln (1 + \frac{8}{x})} (- \frac{8 x^{- 1}}{1 + \frac{8}{x}} + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 7

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

$e^{x \ln (1 + \frac{8}{x})} (- \frac{8}{(1 + \frac{8}{x}) x} + \ln (1 + \frac{8}{x}) \frac{d}{d x} [x])$

Step 8

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

$e^{x \ln (1 + \frac{8}{x})} (- \frac{8}{(1 + \frac{8}{x}) x} + \ln (1 + \frac{8}{x}) \cdot 1)$

Step 9

Multiply $\ln (1 + \frac{8}{x})$ by $1$ .

$e^{x \ln (1 + \frac{8}{x})} (- \frac{8}{(1 + \frac{8}{x}) x} + \ln (1 + \frac{8}{x}))$

Step 10

To write $\ln (1 + \frac{8}{x})$ as a fraction with a common denominator, multiply by $\frac{(1 + \frac{8}{x}) x}{(1 + \frac{8}{x}) x}$ .

$e^{x \ln (1 + \frac{8}{x})} (- \frac{8}{(1 + \frac{8}{x}) x} + \ln (1 + \frac{8}{x}) \cdot \frac{(1 + \frac{8}{x}) x}{(1 + \frac{8}{x}) x})$

Step 11

Combine $\ln (1 + \frac{8}{x})$ and $\frac{(1 + \frac{8}{x}) x}{(1 + \frac{8}{x}) x}$ .

$e^{x \ln (1 + \frac{8}{x})} (- \frac{8}{(1 + \frac{8}{x}) x} + \frac{\ln (1 + \frac{8}{x}) ((1 + \frac{8}{x}) x)}{(1 + \frac{8}{x}) x})$

Step 12

Combine the numerators over the common denominator.

$e^{x \ln (1 + \frac{8}{x})} \frac{- 8 + \ln (1 + \frac{8}{x}) ((1 + \frac{8}{x}) x)}{(1 + \frac{8}{x}) x}$

Step 13

Combine $e^{x \ln (1 + \frac{8}{x})}$ and $\frac{- 8 + \ln (1 + \frac{8}{x}) ((1 + \frac{8}{x}) x)}{(1 + \frac{8}{x}) x}$ .

$\frac{e^{x \ln (1 + \frac{8}{x})} (- 8 + \ln (1 + \frac{8}{x}) ((1 + \frac{8}{x}) x))}{(1 + \frac{8}{x}) x}$

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

$\frac{e^{x \ln (1 + \frac{8}{x})} (- 8 + \ln (1 + \frac{8}{x}) (1 x + \frac{8}{x} x))}{(1 + \frac{8}{x}) x}$

Step 14.2

Apply the distributive property.

$\frac{e^{x \ln (1 + \frac{8}{x})} (- 8 + \ln (1 + \frac{8}{x}) (1 x + \frac{8}{x} x))}{1 x + \frac{8}{x} x}$

Step 14.3

Simplify the numerator.

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Step 14.3.1

Simplify each term.

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Step 14.3.1.1

Simplify each term.

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Step 14.3.1.1.1

Multiply $x$ by $1$ .

$\frac{e^{x \ln (1 + \frac{8}{x})} (- 8 + \ln (1 + \frac{8}{x}) (x + \frac{8}{x} x))}{1 x + \frac{8}{x} x}$

Step 14.3.1.1.2

Cancel the common factor of $x$ .

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Step 14.3.1.1.2.1

Cancel the common factor.

$\frac{e^{x \ln (1 + \frac{8}{x})} (- 8 + \ln (1 + \frac{8}{x}) (x + \frac{8}{x} x))}{1 x + \frac{8}{x} x}$

Step 14.3.1.1.2.2

Rewrite the expression.

$\frac{e^{x \ln (1 + \frac{8}{x})} (- 8 + \ln (1 + \frac{8}{x}) (x + 8))}{1 x + \frac{8}{x} x}$

Step 14.3.1.2

Apply the distributive property.

$\frac{e^{x \ln (1 + \frac{8}{x})} (- 8 + \ln (1 + \frac{8}{x}) x + \ln (1 + \frac{8}{x}) \cdot 8)}{1 x + \frac{8}{x} x}$

Step 14.3.1.3

Multiply $\ln (1 + \frac{8}{x}) \cdot 8$ .

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Step 14.3.1.3.1

Reorder $\ln (1 + \frac{8}{x})$ and $8$ .

$\frac{e^{x \ln (1 + \frac{8}{x})} (- 8 + \ln (1 + \frac{8}{x}) x + 8 \cdot \ln (1 + \frac{8}{x}))}{1 x + \frac{8}{x} x}$

Step 14.3.1.3.2

Simplify $8 \cdot \ln (1 + \frac{8}{x})$ by moving $8$ inside the logarithm.

$\frac{e^{x \ln (1 + \frac{8}{x})} (- 8 + \ln (1 + \frac{8}{x}) x + \ln ({(1 + \frac{8}{x})}^{8}))}{1 x + \frac{8}{x} x}$

Step 14.3.2

Apply the distributive property.

$\frac{e^{x \ln (1 + \frac{8}{x})} \cdot - 8 + e^{x \ln (1 + \frac{8}{x})} (\ln (1 + \frac{8}{x}) x) + e^{x \ln (1 + \frac{8}{x})} \ln ({(1 + \frac{8}{x})}^{8})}{1 x + \frac{8}{x} x}$

Step 14.3.3

Move $- 8$ to the left of $e^{x \ln (1 + \frac{8}{x})}$ .

$\frac{- 8 \cdot e^{x \ln (1 + \frac{8}{x})} + e^{x \ln (1 + \frac{8}{x})} (\ln (1 + \frac{8}{x}) x) + e^{x \ln (1 + \frac{8}{x})} \ln ({(1 + \frac{8}{x})}^{8})}{1 x + \frac{8}{x} x}$

Step 14.3.4

Reorder factors in $- 8 e^{x \ln (1 + \frac{8}{x})} + e^{x \ln (1 + \frac{8}{x})} (\ln (1 + \frac{8}{x}) x) + e^{x \ln (1 + \frac{8}{x})} \ln ({(1 + \frac{8}{x})}^{8})$ .

$\frac{- 8 e^{x \ln (1 + \frac{8}{x})} + x e^{x \ln (1 + \frac{8}{x})} \ln (1 + \frac{8}{x}) + e^{x \ln (1 + \frac{8}{x})} \ln ({(1 + \frac{8}{x})}^{8})}{1 x + \frac{8}{x} x}$

Step 14.4

Combine terms.

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Step 14.4.1

Multiply $x$ by $1$ .

$\frac{- 8 e^{x \ln (1 + \frac{8}{x})} + x e^{x \ln (1 + \frac{8}{x})} \ln (1 + \frac{8}{x}) + e^{x \ln (1 + \frac{8}{x})} \ln ({(1 + \frac{8}{x})}^{8})}{x + \frac{8}{x} x}$

Step 14.4.2

Combine $\frac{8}{x}$ and $x$ .

$\frac{- 8 e^{x \ln (1 + \frac{8}{x})} + x e^{x \ln (1 + \frac{8}{x})} \ln (1 + \frac{8}{x}) + e^{x \ln (1 + \frac{8}{x})} \ln ({(1 + \frac{8}{x})}^{8})}{x + \frac{8 x}{x}}$

Step 14.4.3

Cancel the common factor of $x$ .

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Step 14.4.3.1

Cancel the common factor.

$\frac{- 8 e^{x \ln (1 + \frac{8}{x})} + x e^{x \ln (1 + \frac{8}{x})} \ln (1 + \frac{8}{x}) + e^{x \ln (1 + \frac{8}{x})} \ln ({(1 + \frac{8}{x})}^{8})}{x + \frac{8 x}{x}}$

Step 14.4.3.2

Divide $8$ by $1$ .

$\frac{- 8 e^{x \ln (1 + \frac{8}{x})} + x e^{x \ln (1 + \frac{8}{x})} \ln (1 + \frac{8}{x}) + e^{x \ln (1 + \frac{8}{x})} \ln ({(1 + \frac{8}{x})}^{8})}{x + 8}$

Step 14.5

Reorder terms.

$\frac{e^{x \ln (1 + \frac{8}{x})} x \ln (1 + \frac{8}{x}) + e^{x \ln (1 + \frac{8}{x})} \ln ({(1 + \frac{8}{x})}^{8}) - 8 e^{x \ln (1 + \frac{8}{x})}}{x + 8}$