Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ 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 $g (x) = x - 1$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $1$ and $0$ .

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

Step 1.2.4.2

Multiply $x - 1$ by $1$ .

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

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

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

Step 1.2.8

Simplify the expression.

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Step 1.2.8.1

Add $1$ and $0$ .

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

Step 1.2.8.2

Multiply $- 1$ by $1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Simplify the numerator.

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Step 1.3.2.1

Combine the opposite terms in $x - 1 - x - 1 \cdot 1$ .

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Step 1.3.2.1.1

Subtract $x$ from $x$ .

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

Step 1.3.2.1.2

Subtract $1$ from $0$ .

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

Step 1.3.2.2

Multiply $- 1$ by $1$ .

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

Step 1.3.2.3

Subtract $1$ from $- 1$ .

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

Step 1.3.3

Move the negative in front of the fraction.

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

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 $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2}{{(x - 1)}^{2}}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{1}{{(x - 1)}^{2}}]$ .

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

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

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

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

Step 2.1.2.2

Multiply the exponents in ${({(x - 1)}^{2})}^{- 1}$ .

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Step 2.1.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $x - 1$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 2$ by $- 2$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Simplify the expression.

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Step 2.3.5.1

Add $1$ and $0$ .

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

Step 2.3.5.2

Multiply $4$ by $1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 3

The second derivative of $f (x)$ with respect to $x$ is $\frac{4}{{(x - 1)}^{3}}$ .

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