Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Add $2 x$ and $0$ .

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

Step 1.3

Raise $x$ to the power of $1$ .

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

Step 1.4

Raise $x$ to the power of $1$ .

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

Step 1.5

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

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

Step 1.6

Add $1$ and $1$ .

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

Step 1.7

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

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

Step 1.8

Multiply $- 1$ by $1$ .

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

Step 1.9

Simplify.

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Step 1.9.1

Apply the distributive property.

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

Step 1.9.2

Simplify the numerator.

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Step 1.9.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.9.2.2

Subtract $x^{2}$ from $2 x^{2}$ .

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

Step 2

Find the second derivative.

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Step 2.1

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

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.

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

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{2}{x^{3}}$ .

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