Calculus Examples

$y = (x^{2} + 1) (x + 5 + \frac{1}{x})$

Step 1

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 $f (x) = x^{2} + 1$ and $g (x) = x + 5 + \frac{1}{x}$ .

$(x^{2} + 1) \frac{d}{d x} [x + 5 + \frac{1}{x}] + (x + 5 + \frac{1}{x}) \frac{d}{d x} [x^{2} + 1]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x + 5 + \frac{1}{x}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5] + \frac{d}{d x} [\frac{1}{x}]$ .

$(x^{2} + 1) (\frac{d}{d x} [x] + \frac{d}{d x} [5] + \frac{d}{d x} [\frac{1}{x}]) + (x + 5 + \frac{1}{x}) \frac{d}{d x} [x^{2} + 1]$

Step 2.2

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

$(x^{2} + 1) (1 + \frac{d}{d x} [5] + \frac{d}{d x} [\frac{1}{x}]) + (x + 5 + \frac{1}{x}) \frac{d}{d x} [x^{2} + 1]$

Step 2.3

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$(x^{2} + 1) (1 + 0 + \frac{d}{d x} [\frac{1}{x}]) + (x + 5 + \frac{1}{x}) \frac{d}{d x} [x^{2} + 1]$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $1$ and $0$ .

$(x^{2} + 1) (1 + \frac{d}{d x} [\frac{1}{x}]) + (x + 5 + \frac{1}{x}) \frac{d}{d x} [x^{2} + 1]$

Step 2.4.2

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

$(x^{2} + 1) (1 + \frac{d}{d x} [x^{- 1}]) + (x + 5 + \frac{1}{x}) \frac{d}{d x} [x^{2} + 1]$

Step 2.5

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

$(x^{2} + 1) (1 - x^{- 2}) + (x + 5 + \frac{1}{x}) \frac{d}{d x} [x^{2} + 1]$

Step 2.6

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

$(x^{2} + 1) (1 - x^{- 2}) + (x + 5 + \frac{1}{x}) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])$

Step 2.7

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

$(x^{2} + 1) (1 - x^{- 2}) + (x + 5 + \frac{1}{x}) (2 x + \frac{d}{d x} [1])$

Step 2.8

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

$(x^{2} + 1) (1 - x^{- 2}) + (x + 5 + \frac{1}{x}) (2 x + 0)$

Step 2.9

Simplify the expression.

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Step 2.9.1

Add $2 x$ and $0$ .

$(x^{2} + 1) (1 - x^{- 2}) + (x + 5 + \frac{1}{x}) (2 x)$

Step 2.9.2

Move $2$ to the left of $x + 5 + \frac{1}{x}$ .

$(x^{2} + 1) (1 - x^{- 2}) + 2 \cdot (x + 5 + \frac{1}{x}) x$

Step 3

Simplify.

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Step 3.1

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

$(x^{2} + 1) (1 - \frac{1}{x^{2}}) + 2 (x + 5 + \frac{1}{x}) x$

Step 3.2

Apply the distributive property.

$(x^{2} + 1) (1 - \frac{1}{x^{2}}) + (2 x + 2 \cdot 5 + 2 \frac{1}{x}) x$

Step 3.3

Apply the distributive property.

$(x^{2} + 1) (1 - \frac{1}{x^{2}}) + 2 x \cdot x + 2 \cdot 5 x + 2 \frac{1}{x} x$

Step 3.4

Combine terms.

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Step 3.4.1

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

$(x^{2} + 1) (1 - \frac{1}{x^{2}}) + 2 (x^{1} x) + 2 \cdot 5 x + 2 \frac{1}{x} x$

Step 3.4.2

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

$(x^{2} + 1) (1 - \frac{1}{x^{2}}) + 2 (x^{1} x^{1}) + 2 \cdot 5 x + 2 \frac{1}{x} x$

Step 3.4.3

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

$(x^{2} + 1) (1 - \frac{1}{x^{2}}) + 2 x^{1 + 1} + 2 \cdot 5 x + 2 \frac{1}{x} x$

Step 3.4.4

Add $1$ and $1$ .

$(x^{2} + 1) (1 - \frac{1}{x^{2}}) + 2 x^{2} + 2 \cdot 5 x + 2 \frac{1}{x} x$

Step 3.4.5

Multiply $2$ by $5$ .

$(x^{2} + 1) (1 - \frac{1}{x^{2}}) + 2 x^{2} + 10 x + 2 \frac{1}{x} x$

Step 3.4.6

Combine $2$ and $\frac{1}{x}$ .

$(x^{2} + 1) (1 - \frac{1}{x^{2}}) + 2 x^{2} + 10 x + \frac{2}{x} x$

Step 3.4.7

Combine $\frac{2}{x}$ and $x$ .

$(x^{2} + 1) (1 - \frac{1}{x^{2}}) + 2 x^{2} + 10 x + \frac{2 x}{x}$

Step 3.4.8

Cancel the common factor of $x$ .

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Step 3.4.8.1

Cancel the common factor.

$(x^{2} + 1) (1 - \frac{1}{x^{2}}) + 2 x^{2} + 10 x + \frac{2 x}{x}$

Step 3.4.8.2

Divide $2$ by $1$ .

$(x^{2} + 1) (1 - \frac{1}{x^{2}}) + 2 x^{2} + 10 x + 2$

Step 3.5

Reorder terms.

$2 x^{2} + (1 - \frac{1}{x^{2}}) (x^{2} + 1) + 10 x + 2$

Step 3.6

Simplify each term.

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Step 3.6.1

Expand $(1 - \frac{1}{x^{2}}) (x^{2} + 1)$ using the FOIL Method.

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Step 3.6.1.1

Apply the distributive property.

$2 x^{2} + 1 (x^{2} + 1) - \frac{1}{x^{2}} (x^{2} + 1) + 10 x + 2$

Step 3.6.1.2

Apply the distributive property.

$2 x^{2} + 1 x^{2} + 1 \cdot 1 - \frac{1}{x^{2}} (x^{2} + 1) + 10 x + 2$

Step 3.6.1.3

Apply the distributive property.

$2 x^{2} + 1 x^{2} + 1 \cdot 1 - \frac{1}{x^{2}} x^{2} - \frac{1}{x^{2}} \cdot 1 + 10 x + 2$

Step 3.6.2

Simplify and combine like terms.

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Step 3.6.2.1

Simplify each term.

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Step 3.6.2.1.1

Multiply $x^{2}$ by $1$ .

$2 x^{2} + x^{2} + 1 \cdot 1 - \frac{1}{x^{2}} x^{2} - \frac{1}{x^{2}} \cdot 1 + 10 x + 2$

Step 3.6.2.1.2

Multiply $1$ by $1$ .

$2 x^{2} + x^{2} + 1 - \frac{1}{x^{2}} x^{2} - \frac{1}{x^{2}} \cdot 1 + 10 x + 2$

Step 3.6.2.1.3

Cancel the common factor of $x^{2}$ .

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Step 3.6.2.1.3.1

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

$2 x^{2} + x^{2} + 1 + \frac{- 1}{x^{2}} x^{2} - \frac{1}{x^{2}} \cdot 1 + 10 x + 2$

Step 3.6.2.1.3.2

Cancel the common factor.

$2 x^{2} + x^{2} + 1 + \frac{- 1}{x^{2}} x^{2} - \frac{1}{x^{2}} \cdot 1 + 10 x + 2$

Step 3.6.2.1.3.3

Rewrite the expression.

$2 x^{2} + x^{2} + 1 - 1 - \frac{1}{x^{2}} \cdot 1 + 10 x + 2$

Step 3.6.2.1.4

Multiply $- 1$ by $1$ .

$2 x^{2} + x^{2} + 1 - 1 - \frac{1}{x^{2}} + 10 x + 2$

Step 3.6.2.2

Subtract $1$ from $1$ .

$2 x^{2} + x^{2} + 0 - \frac{1}{x^{2}} + 10 x + 2$

Step 3.6.2.3

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

$2 x^{2} + x^{2} - \frac{1}{x^{2}} + 10 x + 2$

Step 3.7

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

$3 x^{2} - \frac{1}{x^{2}} + 10 x + 2$