Calculus Examples

$y = {(\frac{x + 1}{x - 1})}^{7}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{7}$ and $g (x) = \frac{x + 1}{x - 1}$ .

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Step 1.1.1.1

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

$\frac{d}{d u} [u^{7}] \frac{d}{d x} [\frac{x + 1}{x - 1}]$

Step 1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 7$ .

$7 u^{6} \frac{d}{d x} [\frac{x + 1}{x - 1}]$

Step 1.1.1.3

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

$7 {(\frac{x + 1}{x - 1})}^{6} \frac{d}{d x} [\frac{x + 1}{x - 1}]$

Step 1.1.2

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 $f (x) = x + 1$ and $g (x) = x - 1$ .

$7 {(\frac{x + 1}{x - 1})}^{6} \frac{(x - 1) \frac{d}{d x} [x + 1] - (x + 1) \frac{d}{d x} [x - 1]}{{(x - 1)}^{2}}$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$7 {(\frac{x + 1}{x - 1})}^{6} \frac{(x - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [1]) - (x + 1) \frac{d}{d x} [x - 1]}{{(x - 1)}^{2}}$

Step 1.1.3.2

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

$7 {(\frac{x + 1}{x - 1})}^{6} \frac{(x - 1) (1 + \frac{d}{d x} [1]) - (x + 1) \frac{d}{d x} [x - 1]}{{(x - 1)}^{2}}$

Step 1.1.3.3

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

$7 {(\frac{x + 1}{x - 1})}^{6} \frac{(x - 1) (1 + 0) - (x + 1) \frac{d}{d x} [x - 1]}{{(x - 1)}^{2}}$

Step 1.1.3.4

Simplify the expression.

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Step 1.1.3.4.1

Add $1$ and $0$ .

$7 {(\frac{x + 1}{x - 1})}^{6} \frac{(x - 1) \cdot 1 - (x + 1) \frac{d}{d x} [x - 1]}{{(x - 1)}^{2}}$

Step 1.1.3.4.2

Multiply $x - 1$ by $1$ .

$7 {(\frac{x + 1}{x - 1})}^{6} \frac{x - 1 - (x + 1) \frac{d}{d x} [x - 1]}{{(x - 1)}^{2}}$

Step 1.1.3.5

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

$7 {(\frac{x + 1}{x - 1})}^{6} \frac{x - 1 - (x + 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])}{{(x - 1)}^{2}}$

Step 1.1.3.6

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

$7 {(\frac{x + 1}{x - 1})}^{6} \frac{x - 1 - (x + 1) (1 + \frac{d}{d x} [- 1])}{{(x - 1)}^{2}}$

Step 1.1.3.7

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

$7 {(\frac{x + 1}{x - 1})}^{6} \frac{x - 1 - (x + 1) (1 + 0)}{{(x - 1)}^{2}}$

Step 1.1.3.8

Combine fractions.

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Step 1.1.3.8.1

Add $1$ and $0$ .

$7 {(\frac{x + 1}{x - 1})}^{6} \frac{x - 1 - (x + 1) \cdot 1}{{(x - 1)}^{2}}$

Step 1.1.3.8.2

Multiply $- 1$ by $1$ .

$7 {(\frac{x + 1}{x - 1})}^{6} \frac{x - 1 - (x + 1)}{{(x - 1)}^{2}}$

Step 1.1.3.8.3

Combine $\frac{x - 1 - (x + 1)}{{(x - 1)}^{2}}$ and $7$ .

$\frac{(x - 1 - (x + 1)) \cdot 7}{{(x - 1)}^{2}} {(\frac{x + 1}{x - 1})}^{6}$

Step 1.1.3.8.4

Move $7$ to the left of $x - 1 - (x + 1)$ .

$\frac{7 \cdot (x - 1 - (x + 1))}{{(x - 1)}^{2}} {(\frac{x + 1}{x - 1})}^{6}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

$\frac{7 (x - 1 - (x + 1))}{{(x - 1)}^{2}} \cdot \frac{{(x + 1)}^{6}}{{(x - 1)}^{6}}$

Step 1.1.4.2

Apply the distributive property.

$\frac{7 (x - 1 - x - 1 \cdot 1)}{{(x - 1)}^{2}} \cdot \frac{{(x + 1)}^{6}}{{(x - 1)}^{6}}$

Step 1.1.4.3

Apply the distributive property.

$\frac{7 x + 7 \cdot - 1 + 7 (- x) + 7 (- 1 \cdot 1)}{{(x - 1)}^{2}} \cdot \frac{{(x + 1)}^{6}}{{(x - 1)}^{6}}$

Step 1.1.4.4

Combine terms.

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Step 1.1.4.4.1

Multiply $7$ by $- 1$ .

$\frac{7 x - 7 + 7 (- x) + 7 (- 1 \cdot 1)}{{(x - 1)}^{2}} \cdot \frac{{(x + 1)}^{6}}{{(x - 1)}^{6}}$

Step 1.1.4.4.2

Multiply $- 1$ by $7$ .

$\frac{7 x - 7 - 7 x + 7 (- 1 \cdot 1)}{{(x - 1)}^{2}} \cdot \frac{{(x + 1)}^{6}}{{(x - 1)}^{6}}$

Step 1.1.4.4.3

Multiply $- 1$ by $1$ .

$\frac{7 x - 7 - 7 x + 7 \cdot - 1}{{(x - 1)}^{2}} \cdot \frac{{(x + 1)}^{6}}{{(x - 1)}^{6}}$

Step 1.1.4.4.4

Multiply $7$ by $- 1$ .

$\frac{7 x - 7 - 7 x - 7}{{(x - 1)}^{2}} \cdot \frac{{(x + 1)}^{6}}{{(x - 1)}^{6}}$

Step 1.1.4.4.5

Subtract $7 x$ from $7 x$ .

$\frac{0 - 7 - 7}{{(x - 1)}^{2}} \cdot \frac{{(x + 1)}^{6}}{{(x - 1)}^{6}}$

Step 1.1.4.4.6

Subtract $7$ from $0$ .

$\frac{- 7 - 7}{{(x - 1)}^{2}} \cdot \frac{{(x + 1)}^{6}}{{(x - 1)}^{6}}$

Step 1.1.4.4.7

Subtract $7$ from $- 7$ .

$\frac{- 14}{{(x - 1)}^{2}} \cdot \frac{{(x + 1)}^{6}}{{(x - 1)}^{6}}$

Step 1.1.4.4.8

Move the negative in front of the fraction.

$- \frac{14}{{(x - 1)}^{2}} \cdot \frac{{(x + 1)}^{6}}{{(x - 1)}^{6}}$

Step 1.1.4.4.9

Multiply $\frac{{(x + 1)}^{6}}{{(x - 1)}^{6}}$ by $\frac{14}{{(x - 1)}^{2}}$ .

$- \frac{{(x + 1)}^{6} \cdot 14}{{(x - 1)}^{6} {(x - 1)}^{2}}$

Step 1.1.4.4.10

Multiply ${(x - 1)}^{6}$ by ${(x - 1)}^{2}$ by adding the exponents.

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Step 1.1.4.4.10.1

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

$- \frac{{(x + 1)}^{6} \cdot 14}{{(x - 1)}^{6 + 2}}$

Step 1.1.4.4.10.2

Add $6$ and $2$ .

$- \frac{{(x + 1)}^{6} \cdot 14}{{(x - 1)}^{8}}$

Step 1.1.4.4.11

Move $14$ to the left of ${(x + 1)}^{6}$ .

$f' (x) = - \frac{14 {(x + 1)}^{6}}{{(x - 1)}^{8}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{14 {(x + 1)}^{6}}{{(x - 1)}^{8}}$ .

$- \frac{14 {(x + 1)}^{6}}{{(x - 1)}^{8}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{14 {(x + 1)}^{6}}{{(x - 1)}^{8}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{14 {(x + 1)}^{6}}{{(x - 1)}^{8}} = 0$

Step 2.2

Set the numerator equal to zero.

$14 {(x + 1)}^{6} = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $14 {(x + 1)}^{6} = 0$ by $14$ and simplify.

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Step 2.3.1.1

Divide each term in $14 {(x + 1)}^{6} = 0$ by $14$ .

$\frac{14 {(x + 1)}^{6}}{14} = \frac{0}{14}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $14$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{14 {(x + 1)}^{6}}{14} = \frac{0}{14}$

Step 2.3.1.2.1.2

Divide ${(x + 1)}^{6}$ by $1$ .

${(x + 1)}^{6} = \frac{0}{14}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $14$ .

${(x + 1)}^{6} = 0$

Step 2.3.2

Set the $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.3.3

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{14 {(x + 1)}^{6}}{{(x - 1)}^{8}}$ equal to $0$ to find where the expression is undefined.

${(x - 1)}^{8} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4

Evaluate ${(\frac{x + 1}{x - 1})}^{7}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 1$ .

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Step 4.1.1

Substitute $- 1$ for $x$ .

${(\frac{(- 1) + 1}{(- 1) - 1})}^{7}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Add $- 1$ and $1$ .

${(\frac{0}{- 1 - 1})}^{7}$

Step 4.1.2.2

Subtract $1$ from $- 1$ .

${(\frac{0}{- 2})}^{7}$

Step 4.1.2.3

Divide $0$ by $- 2$ .

$0^{7}$

Step 4.1.2.4

Raising $0$ to any positive power yields $0$ .

$0$

Step 4.2

Evaluate at $x = 1$ .

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Step 4.2.1

Substitute $1$ for $x$ .

${(\frac{(1) + 1}{(1) - 1})}^{7}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Subtract $1$ from $1$ .

${(\frac{(1) + 1}{0})}^{7}$

Step 4.2.2.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.3

List all of the points.

$(- 1, 0)$

Step 5