Calculus Examples

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

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

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^{2} - 1

$f (x) = x^{2} - 1$ and

g (x) = x

$g (x) = x$ .

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

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of

x^{2} - 1

$x^{2} - 1$ with respect to

x

$x$ is

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

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

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

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

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 = 2

$n = 2$ .

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

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

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

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

Step 1.2.4

Add

2 x

$2 x$ and

0

$0$ .

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

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

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

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

Step 1.3

Raise

x

$x$ to the power of

1

$1$ .

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

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

Step 1.4

Raise

x

$x$ to the power of

1

$1$ .

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

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

Step 1.5

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 1.6

Add

1

$1$ and

1

$1$ .

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

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

Step 1.7

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

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

Step 1.8

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 1.9

Simplify.

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Step 1.9.1

Apply the distributive property.

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

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

Step 1.9.2

Simplify the numerator.

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Step 1.9.2.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 1.9.2.2

Subtract

x^{2}

$x^{2}$ from

2 x^{2}

$2 x^{2}$ .

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

Step 2

Find the second derivative.

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Step 2.1

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

$f (x) = x^{2} + 1$ and

$g (x) = x^{2}$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

Multiply the exponents in

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

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Step 2.2.1.1

Apply the power rule and multiply exponents,

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

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

Step 2.2.1.2

Multiply

$2$ by

$2$ .

$\frac{x^{2} \frac{d}{d x} [x^{2} + 1] - (x^{2} + 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.2

By the Sum Rule, the derivative of

$x^{2} + 1$ with respect to

$x$ is

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

$\frac{x^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]) - (x^{2} + 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.3

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 2$ .

$\frac{x^{2} (2 x + \frac{d}{d x} [1]) - (x^{2} + 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.4

Since

$1$ is constant with respect to

$x$ , the derivative of

$1$ with respect to

$x$ is

$0$ .

$\frac{x^{2} (2 x + 0) - (x^{2} + 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.5

Add

$2 x$ and

$0$ .

$\frac{x^{2} (2 x) - (x^{2} + 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.3

Multiply

$x^{2}$ by

$x$ by adding the exponents.

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Step 2.3.1

Move

$x$ .

$\frac{x \cdot x^{2} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.3.2

Multiply

$x$ by

$x^{2}$ .

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Step 2.3.2.1

Raise

$x$ to the power of

$1$ .

$\frac{x^{1} x^{2} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.3.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 2} \cdot 2 - (x^{2} + 1) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.3.3

Add

$1$ and

$2$ .

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

Step 2.4

Move

$2$ to the left of

$x^{3}$ .

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

Step 2.5

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 2$ .

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

Step 2.6

Multiply

$2$ by

$- 1$ .

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

Step 2.7

Simplify.

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Step 2.7.1

Apply the distributive property.

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

Step 2.7.2

Apply the distributive property.

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

Step 2.7.3

Simplify the numerator.

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Step 2.7.3.1

Simplify each term.

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Step 2.7.3.1.1

Multiply

$x^{2}$ by

$x$ by adding the exponents.

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Step 2.7.3.1.1.1

Move

$x$ .

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

Step 2.7.3.1.1.2

Multiply

$x$ by

$x^{2}$ .

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Step 2.7.3.1.1.2.1

Raise

$x$ to the power of

$1$ .

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

Step 2.7.3.1.1.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.7.3.1.1.3

Add

$1$ and

$2$ .

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

Step 2.7.3.1.2

Multiply

$- 2$ by

$1$ .

$\frac{2 x^{3} - 2 x^{3} - 2 x}{x^{4}}$

Step 2.7.3.2

Subtract

$2 x^{3}$ from

$2 x^{3}$ .

$\frac{0 - 2 x}{x^{4}}$

Step 2.7.3.3

Subtract

$2 x$ from

$0$ .

$\frac{- 2 x}{x^{4}}$

Step 2.7.4

Combine terms.

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Step 2.7.4.1

Cancel the common factor of

$x$ and

$x^{4}$ .

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Step 2.7.4.1.1

Factor

$x$ out of

$- 2 x$ .

$\frac{x \cdot - 2}{x^{4}}$

Step 2.7.4.1.2

Cancel the common factors.

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Step 2.7.4.1.2.1

Factor

$x$ out of

$x^{4}$ .

$\frac{x \cdot - 2}{x \cdot x^{3}}$

Step 2.7.4.1.2.2

Cancel the common factor.

$\frac{x \cdot - 2}{x \cdot x^{3}}$

Step 2.7.4.1.2.3

Rewrite the expression.

$\frac{- 2}{x^{3}}$

Step 2.7.4.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{2}{x^{3}}$