Calculus Examples

f (x) = \frac{x + 1}{x - 1}

$f (x) = \frac{x + 1}{x - 1}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Quotient Rule which states that

\frac{d}{d x} [\frac{f (x)}{g (x)}]

$\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is

\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}

$\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where

f (x) = x + 1

$f (x) = x + 1$ and

g (x) = x - 1

$g (x) = x - 1$ .

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

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of

x + 1

$x + 1$ with respect to

x

$x$ is

\frac{d}{d x} [x] + \frac{d}{d x} [1]

$\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

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

Step 1.2.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

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

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

Step 1.2.3

Since

1

$1$ is constant with respect to

x

$x$ , the derivative of

1

$1$ with respect to

x

$x$ is

0

$0$ .

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

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

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add

1

$1$ and

0

$0$ .

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

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

Step 1.2.4.2

Multiply

x - 1

$x - 1$ by

1

$1$ .

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

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

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

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

Step 1.2.5

By the Sum Rule, the derivative of

x - 1

$x - 1$ with respect to

x

$x$ is

\frac{d}{d x} [x] + \frac{d}{d x} [- 1]

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

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

Step 1.2.6

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

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

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

Step 1.2.7

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- 1

$- 1$ with respect to

x

$x$ is

0

$0$ .

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

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

Step 1.2.8

Simplify the expression.

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Step 1.2.8.1

Add

1

$1$ and

0

$0$ .

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

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

Step 1.2.8.2

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

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

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

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

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

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

Step 1.3.2

Simplify the numerator.

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Step 1.3.2.1

Combine the opposite terms in

x - 1 - x - 1 \cdot 1

$x - 1 - x - 1 \cdot 1$ .

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Step 1.3.2.1.1

Subtract

x

$x$ from

x

$x$ .

\frac{0 - 1 - 1 \cdot 1}{{(x - 1)}^{2}}

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

Step 1.3.2.1.2

Subtract

1

$1$ from

0

$0$ .

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

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

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

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

Step 1.3.2.2

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 1.3.2.3

Subtract

1

$1$ from

- 1

$- 1$ .

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

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

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

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

Step 1.3.3

Move the negative in front of the fraction.

f' (x) = - \frac{2}{{(x - 1)}^{2}}

$f' (x) = - \frac{2}{{(x - 1)}^{2}}$

f' (x) = - \frac{2}{{(x - 1)}^{2}}

$f' (x) = - \frac{2}{{(x - 1)}^{2}}$

f' (x) = - \frac{2}{{(x - 1)}^{2}}

$f' (x) = - \frac{2}{{(x - 1)}^{2}}$

Step 2

Find the second derivative.

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Step 2.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1

Since

- 2

$- 2$ is constant with respect to

x

$x$ , the derivative of

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

$- \frac{2}{{(x - 1)}^{2}}$ with respect to

x

$x$ is

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

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

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

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

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

Rewrite

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

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

{({(x - 1)}^{2})}^{- 1}

${({(x - 1)}^{2})}^{- 1}$ .

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

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

Step 2.1.2.2

Multiply the exponents in

{({(x - 1)}^{2})}^{- 1}

${({(x - 1)}^{2})}^{- 1}$ .

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Step 2.1.2.2.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 2.1.2.2.2

Multiply

2

$2$ by

- 1

$- 1$ .

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

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

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

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

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

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

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

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

Step 2.2

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

f (x) = x^{- 2}

$f (x) = x^{- 2}$ and

g (x) = x - 1

$g (x) = x - 1$ .

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Step 2.2.1

To apply the Chain Rule, set

u

$u$ as

x - 1

$x - 1$ .

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

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

Step 2.2.2

Differentiate using the Power Rule which states that

\frac{d}{d u} [u^{n}]

$\frac{d}{d u} [u^{n}]$ is

n u^{n - 1}

$n u^{n - 1}$ where

n = - 2

$n = - 2$ .

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

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

Step 2.2.3

Replace all occurrences of

u

$u$ with

x - 1

$x - 1$ .

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

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

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

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

Step 2.3

Differentiate.

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Step 2.3.1

Multiply

- 2

$- 2$ by

- 2

$- 2$ .

4 ({(x - 1)}^{- 3} \frac{d}{d x} [x - 1])

$4 ({(x - 1)}^{- 3} \frac{d}{d x} [x - 1])$

Step 2.3.2

By the Sum Rule, the derivative of

x - 1

$x - 1$ with respect to

x

$x$ is

\frac{d}{d x} [x] + \frac{d}{d x} [- 1]

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

$4 {(x - 1)}^{- 3} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 2.3.3

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

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

$4 {(x - 1)}^{- 3} (1 + \frac{d}{d x} [- 1])$

Step 2.3.4

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- 1

$- 1$ with respect to

x

$x$ is

0

$0$ .

4 {(x - 1)}^{- 3} (1 + 0)

$4 {(x - 1)}^{- 3} (1 + 0)$

Step 2.3.5

Simplify the expression.

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Step 2.3.5.1

Add

1

$1$ and

0

$0$ .

4 {(x - 1)}^{- 3} \cdot 1

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

Step 2.3.5.2

Multiply

4

$4$ by

1

$1$ .

4 {(x - 1)}^{- 3}

$4 {(x - 1)}^{- 3}$

4 {(x - 1)}^{- 3}

$4 {(x - 1)}^{- 3}$

4 {(x - 1)}^{- 3}

$4 {(x - 1)}^{- 3}$

Step 2.4

Simplify.

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Step 2.4.1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 2.4.2

Combine

4

$4$ and

\frac{1}{{(x - 1)}^{3}}

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

f'' (x) = \frac{4}{{(x - 1)}^{3}}

$f'' (x) = \frac{4}{{(x - 1)}^{3}}$

f'' (x) = \frac{4}{{(x - 1)}^{3}}

$f'' (x) = \frac{4}{{(x - 1)}^{3}}$

f'' (x) = \frac{4}{{(x - 1)}^{3}}

$f'' (x) = \frac{4}{{(x - 1)}^{3}}$

Step 3

The second derivative of

f (x)

$f (x)$ with respect to

x

$x$ is

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

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

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

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