Calculus Examples

$y = {(1 + \frac{1}{x})}^{3}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 1 + \frac{1}{x}$ .

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Step 1.1

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

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [1 + \frac{1}{x}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$3 u^{2} \frac{d}{d x} [1 + \frac{1}{x}]$

Step 1.3

Replace all occurrences of $u$ with $1 + \frac{1}{x}$ .

$3 {(1 + \frac{1}{x})}^{2} \frac{d}{d x} [1 + \frac{1}{x}]$

Step 2

Differentiate.

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Step 2.1

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

$3 {(1 + \frac{1}{x})}^{2} (\frac{d}{d x} [1] + \frac{d}{d x} [\frac{1}{x}])$

Step 2.2

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

$3 {(1 + \frac{1}{x})}^{2} (0 + \frac{d}{d x} [\frac{1}{x}])$

Step 2.3

Simplify the expression.

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Step 2.3.1

Add $0$ and $\frac{d}{d x} [\frac{1}{x}]$ .

$3 {(1 + \frac{1}{x})}^{2} \frac{d}{d x} [\frac{1}{x}]$

Step 2.3.2

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

$3 {(1 + \frac{1}{x})}^{2} \frac{d}{d x} [x^{- 1}]$

Step 2.4

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

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

Step 2.5

Multiply $- 1$ by $3$ .

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

Step 3

Simplify.

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Step 3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.2

Combine terms.

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Step 3.2.1

Combine $\frac{1}{x^{2}}$ and $- 3$ .

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

Step 3.2.2

Combine $\frac{- 3}{x^{2}}$ and ${(1 + \frac{1}{x})}^{2}$ .

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

Step 3.2.3

Move the negative in front of the fraction.

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

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

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

Step 3.3.2

Combine the numerators over the common denominator.

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

Step 3.3.3

Apply the product rule to $\frac{x + 1}{x}$ .

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

Step 3.4

Combine $3$ and $\frac{{(x + 1)}^{2}}{x^{2}}$ .

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

Step 3.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.6

Combine.

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

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

Add $2$ and $2$ .

$- \frac{3 {(x + 1)}^{2} \cdot 1}{x^{4}}$

Step 3.8

Multiply $3 {(x + 1)}^{2}$ by $1$ .

$- \frac{3 {(x + 1)}^{2}}{x^{4}}$