Calculus Examples

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

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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 $g (x) = x$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

$f' (x) = \frac{x (2 x + 0) - (x^{2} + 1) \frac{d}{d x} x}{x^{2}}$

Step 1.1.2.4

Add $2 x$ and $0$ .

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

Add $1$ and $1$ .

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

Step 1.1.7

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

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

Step 1.1.8

Multiply $- 1$ by $1$ .

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

Step 1.1.9

Simplify.

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Step 1.1.9.1

Apply the distributive property.

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

Step 1.1.9.2

Simplify the numerator.

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Step 1.1.9.2.1

Multiply $- 1$ by $1$ .

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

Step 1.1.9.2.2

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

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

Step 1.1.9.3

Simplify the numerator.

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Step 1.1.9.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.1.9.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 1.2

Find the second derivative.

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Step 1.2.1

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{x^{2} \frac{d}{d x} ((x + 1) (x - 1)) - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.3

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

$f'' (x) = \frac{x^{2} ((x + 1) \frac{d}{d x} (x - 1) + (x - 1) \frac{d}{d x} (x + 1)) - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.4

Differentiate.

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Step 1.2.4.1

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

$f'' (x) = \frac{x^{2} ((x + 1) (\frac{d}{d x} (x) + \frac{d}{d x} (- 1)) + (x - 1) \frac{d}{d x} (x + 1)) - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.4.2

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

$f'' (x) = \frac{x^{2} ((x + 1) (1 + \frac{d}{d x} (- 1)) + (x - 1) \frac{d}{d x} (x + 1)) - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.4.3

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

$f'' (x) = \frac{x^{2} ((x + 1) (1 + 0) + (x - 1) \frac{d}{d x} (x + 1)) - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.4.4

Simplify the expression.

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Step 1.2.4.4.1

Add $1$ and $0$ .

$f'' (x) = \frac{x^{2} ((x + 1) \cdot 1 + (x - 1) \frac{d}{d x} (x + 1)) - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.4.4.2

Multiply $x + 1$ by $1$ .

$f'' (x) = \frac{x^{2} (x + 1 + (x - 1) \frac{d}{d x} (x + 1)) - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.4.5

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

$f'' (x) = \frac{x^{2} (x + 1 + (x - 1) (\frac{d}{d x} (x) + \frac{d}{d x} (1))) - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.4.6

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

$f'' (x) = \frac{x^{2} (x + 1 + (x - 1) (1 + \frac{d}{d x} (1))) - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.4.7

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

$f'' (x) = \frac{x^{2} (x + 1 + (x - 1) (1 + 0)) - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.4.8

Simplify by adding terms.

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Step 1.2.4.8.1

Add $1$ and $0$ .

$f'' (x) = \frac{x^{2} (x + 1 + (x - 1) \cdot 1) - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.4.8.2

Multiply $x - 1$ by $1$ .

$f'' (x) = \frac{x^{2} (x + 1 + x - 1) - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.4.8.3

Add $x$ and $x$ .

$f'' (x) = \frac{x^{2} (2 x + 1 - 1) - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.4.8.4

Simplify by subtracting numbers.

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Step 1.2.4.8.4.1

Subtract $1$ from $1$ .

$f'' (x) = \frac{x^{2} (2 x + 0) - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.4.8.4.2

Add $2 x$ and $0$ .

$f'' (x) = \frac{x^{2} (2 x) - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.2.5.1

Move $x$ .

$f'' (x) = \frac{x \cdot x^{2} \cdot 2 - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.5.2

Multiply $x$ by $x^{2}$ .

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Step 1.2.5.2.1

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

$f'' (x) = \frac{x \cdot x^{2} \cdot 2 - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.5.2.2

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

$f'' (x) = \frac{x^{1 + 2} \cdot 2 - (x + 1) (x - 1) \frac{d}{d x} x^{2}}{x^{4}}$

Step 1.2.5.3

Add $1$ and $2$ .

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

Step 1.2.6

Move $2$ to the left of $x^{3}$ .

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

Step 1.2.7

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

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

Step 1.2.8

Multiply $2$ by $- 1$ .

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

Step 1.2.9

Simplify.

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Step 1.2.9.1

Apply the distributive property.

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

Step 1.2.9.2

Simplify the numerator.

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Step 1.2.9.2.1

Simplify each term.

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Step 1.2.9.2.1.1

Multiply $- 2$ by $1$ .

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

Step 1.2.9.2.1.2

Expand $(- 2 x - 2) (x - 1)$ using the FOIL Method.

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Step 1.2.9.2.1.2.1

Apply the distributive property.

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

Step 1.2.9.2.1.2.2

Apply the distributive property.

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

Step 1.2.9.2.1.2.3

Apply the distributive property.

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

Step 1.2.9.2.1.3

Simplify and combine like terms.

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Step 1.2.9.2.1.3.1

Simplify each term.

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Step 1.2.9.2.1.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.9.2.1.3.1.1.1

Move $x$ .

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

Step 1.2.9.2.1.3.1.1.2

Multiply $x$ by $x$ .

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

Step 1.2.9.2.1.3.1.2

Multiply $- 1$ by $- 2$ .

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

Step 1.2.9.2.1.3.1.3

Multiply $- 2$ by $- 1$ .

$f'' (x) = \frac{2 x^{3} + (- 2 x^{2} + 2 x - 2 x + 2) x}{x^{4}}$

Step 1.2.9.2.1.3.2

Subtract $2 x$ from $2 x$ .

$f'' (x) = \frac{2 x^{3} + (- 2 x^{2} + 0 + 2) x}{x^{4}}$

Step 1.2.9.2.1.3.3

Add $- 2 x^{2}$ and $0$ .

$f'' (x) = \frac{2 x^{3} + (- 2 x^{2} + 2) x}{x^{4}}$

Step 1.2.9.2.1.4

Apply the distributive property.

$f'' (x) = \frac{2 x^{3} - 2 x^{2} x + 2 x}{x^{4}}$

Step 1.2.9.2.1.5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.2.9.2.1.5.1

Move $x$ .

$f'' (x) = \frac{2 x^{3} - 2 (x \cdot x^{2}) + 2 x}{x^{4}}$

Step 1.2.9.2.1.5.2

Multiply $x$ by $x^{2}$ .

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Step 1.2.9.2.1.5.2.1

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

$f'' (x) = \frac{2 x^{3} - 2 (x \cdot x^{2}) + 2 x}{x^{4}}$

Step 1.2.9.2.1.5.2.2

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

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

Step 1.2.9.2.1.5.3

Add $1$ and $2$ .

$f'' (x) = \frac{2 x^{3} - 2 x^{3} + 2 x}{x^{4}}$

Step 1.2.9.2.2

Subtract $2 x^{3}$ from $2 x^{3}$ .

$f'' (x) = \frac{0 + 2 x}{x^{4}}$

Step 1.2.9.2.3

Add $0$ and $2 x$ .

$f'' (x) = \frac{2 x}{x^{4}}$

Step 1.2.9.3

Cancel the common factor of $x$ and $x^{4}$ .

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Step 1.2.9.3.1

Factor $x$ out of $2 x$ .

$f'' (x) = \frac{x \cdot 2}{x^{4}}$

Step 1.2.9.3.2

Cancel the common factors.

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Step 1.2.9.3.2.1

Factor $x$ out of $x^{4}$ .

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

Step 1.2.9.3.2.2

Cancel the common factor.

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

Step 1.2.9.3.2.3

Rewrite the expression.

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

Step 1.3

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

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

Step 2

Set the second derivative equal to $0$ then solve the equation $\frac{2}{x^{3}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$\frac{2}{x^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$2 = 0$

Step 2.3

Since $2 \neq 0$ , there are no solutions.

No solution

Step 3

No values found that can make the second derivative equal to $0$ .

No Inflection Points