Calculus Examples

$(\frac{x}{3.2} + \frac{3.2}{x}) (x^{2} + 1)$

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 $f (x) = \frac{x}{3.2} + \frac{3.2}{x}$ and $g (x) = x^{2} + 1$ .

$(\frac{x}{3.2} + \frac{3.2}{x}) \frac{d}{d x} [x^{2} + 1] + (x^{2} + 1) \frac{d}{d x} [\frac{x}{3.2} + \frac{3.2}{x}]$

Step 2

Differentiate.

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Step 2.1

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

$(\frac{x}{3.2} + \frac{3.2}{x}) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]) + (x^{2} + 1) \frac{d}{d x} [\frac{x}{3.2} + \frac{3.2}{x}]$

Step 2.2

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

$(\frac{x}{3.2} + \frac{3.2}{x}) (2 x + \frac{d}{d x} [1]) + (x^{2} + 1) \frac{d}{d x} [\frac{x}{3.2} + \frac{3.2}{x}]$

Step 2.3

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

$(\frac{x}{3.2} + \frac{3.2}{x}) (2 x + 0) + (x^{2} + 1) \frac{d}{d x} [\frac{x}{3.2} + \frac{3.2}{x}]$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $2 x$ and $0$ .

$(\frac{x}{3.2} + \frac{3.2}{x}) (2 x) + (x^{2} + 1) \frac{d}{d x} [\frac{x}{3.2} + \frac{3.2}{x}]$

Step 2.4.2

Move $2$ to the left of $\frac{x}{3.2} + \frac{3.2}{x}$ .

$2 \cdot (\frac{x}{3.2} + \frac{3.2}{x}) x + (x^{2} + 1) \frac{d}{d x} [\frac{x}{3.2} + \frac{3.2}{x}]$

Step 2.5

By the Sum Rule, the derivative of $\frac{x}{3.2} + \frac{3.2}{x}$ with respect to $x$ is $\frac{d}{d x} [\frac{x}{3.2}] + \frac{d}{d x} [\frac{3.2}{x}]$ .

$2 (\frac{x}{3.2} + \frac{3.2}{x}) x + (x^{2} + 1) (\frac{d}{d x} [\frac{x}{3.2}] + \frac{d}{d x} [\frac{3.2}{x}])$

Step 2.6

Since $\frac{1}{3.2}$ is constant with respect to $x$ , the derivative of $\frac{x}{3.2}$ with respect to $x$ is $\frac{1}{3.2} \frac{d}{d x} [x]$ .

$2 (\frac{x}{3.2} + \frac{3.2}{x}) x + (x^{2} + 1) (\frac{1}{3.2} \frac{d}{d x} [x] + \frac{d}{d x} [\frac{3.2}{x}])$

Step 2.7

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

$2 (\frac{x}{3.2} + \frac{3.2}{x}) x + (x^{2} + 1) (\frac{1}{3.2} \cdot 1 + \frac{d}{d x} [\frac{3.2}{x}])$

Step 2.8

Multiply $\frac{1}{3.2}$ by $1$ .

$2 (\frac{x}{3.2} + \frac{3.2}{x}) x + (x^{2} + 1) (\frac{1}{3.2} + \frac{d}{d x} [\frac{3.2}{x}])$

Step 2.9

Since $3.2$ is constant with respect to $x$ , the derivative of $\frac{3.2}{x}$ with respect to $x$ is $3.2 \frac{d}{d x} [\frac{1}{x}]$ .

$2 (\frac{x}{3.2} + \frac{3.2}{x}) x + (x^{2} + 1) (\frac{1}{3.2} + 3.2 \frac{d}{d x} [\frac{1}{x}])$

Step 2.10

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

$2 (\frac{x}{3.2} + \frac{3.2}{x}) x + (x^{2} + 1) (\frac{1}{3.2} + 3.2 \frac{d}{d x} [x^{- 1}])$

Step 2.11

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

$2 (\frac{x}{3.2} + \frac{3.2}{x}) x + (x^{2} + 1) (\frac{1}{3.2} + 3.2 (- x^{- 2}))$

Step 2.12

Multiply $- 1$ by $3.2$ .

$2 (\frac{x}{3.2} + \frac{3.2}{x}) x + (x^{2} + 1) (\frac{1}{3.2} - 3.2 x^{- 2})$

Step 3

Simplify.

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Step 3.1

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

$2 (\frac{x}{3.2} + \frac{3.2}{x}) x + (x^{2} + 1) (\frac{1}{3.2} - 3.2 \frac{1}{x^{2}})$

Step 3.2

Apply the distributive property.

$(2 \frac{x}{3.2} + 2 \frac{3.2}{x}) x + (x^{2} + 1) (\frac{1}{3.2} - 3.2 \frac{1}{x^{2}})$

Step 3.3

Apply the distributive property.

$2 \frac{x}{3.2} x + 2 \frac{3.2}{x} x + (x^{2} + 1) (\frac{1}{3.2} - 3.2 \frac{1}{x^{2}})$

Step 3.4

Combine terms.

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Step 3.4.1

Combine $2$ and $\frac{x}{3.2}$ .

$\frac{2 x}{3.2} x + 2 \frac{3.2}{x} x + (x^{2} + 1) (\frac{1}{3.2} - 3.2 \frac{1}{x^{2}})$

Step 3.4.2

Combine $\frac{2 x}{3.2}$ and $x$ .

$\frac{2 x \cdot x}{3.2} + 2 \frac{3.2}{x} x + (x^{2} + 1) (\frac{1}{3.2} - 3.2 \frac{1}{x^{2}})$

Step 3.4.3

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

$\frac{2 (x^{1} x)}{3.2} + 2 \frac{3.2}{x} x + (x^{2} + 1) (\frac{1}{3.2} - 3.2 \frac{1}{x^{2}})$

Step 3.4.4

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

$\frac{2 (x^{1} x^{1})}{3.2} + 2 \frac{3.2}{x} x + (x^{2} + 1) (\frac{1}{3.2} - 3.2 \frac{1}{x^{2}})$

Step 3.4.5

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

$\frac{2 x^{1 + 1}}{3.2} + 2 \frac{3.2}{x} x + (x^{2} + 1) (\frac{1}{3.2} - 3.2 \frac{1}{x^{2}})$

Step 3.4.6

Add $1$ and $1$ .

$\frac{2 x^{2}}{3.2} + 2 \frac{3.2}{x} x + (x^{2} + 1) (\frac{1}{3.2} - 3.2 \frac{1}{x^{2}})$

Step 3.4.7

Combine $2$ and $\frac{3.2}{x}$ .

$\frac{2 x^{2}}{3.2} + \frac{2 \cdot 3.2}{x} x + (x^{2} + 1) (\frac{1}{3.2} - 3.2 \frac{1}{x^{2}})$

Step 3.4.8

Multiply $2$ by $3.2$ .

$\frac{2 x^{2}}{3.2} + \frac{6.4}{x} x + (x^{2} + 1) (\frac{1}{3.2} - 3.2 \frac{1}{x^{2}})$

Step 3.4.9

Combine $\frac{6.4}{x}$ and $x$ .

$\frac{2 x^{2}}{3.2} + \frac{6.4 x}{x} + (x^{2} + 1) (\frac{1}{3.2} - 3.2 \frac{1}{x^{2}})$

Step 3.4.10

Cancel the common factor of $x$ .

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Step 3.4.10.1

Cancel the common factor.

$\frac{2 x^{2}}{3.2} + \frac{6.4 x}{x} + (x^{2} + 1) (\frac{1}{3.2} - 3.2 \frac{1}{x^{2}})$

Step 3.4.10.2

Divide $6.4$ by $1$ .

$\frac{2 x^{2}}{3.2} + 6.4 + (x^{2} + 1) (\frac{1}{3.2} - 3.2 \frac{1}{x^{2}})$

Step 3.4.11

Combine $- 3.2$ and $\frac{1}{x^{2}}$ .

$\frac{2 x^{2}}{3.2} + 6.4 + (x^{2} + 1) (\frac{1}{3.2} + \frac{- 3.2}{x^{2}})$

Step 3.4.12

Move the negative in front of the fraction.

$\frac{2 x^{2}}{3.2} + 6.4 + (x^{2} + 1) (\frac{1}{3.2} - \frac{3.2}{x^{2}})$

Step 3.5

Reorder terms.

$\frac{2 x^{2}}{3.2} + (x^{2} + 1) (\frac{1}{3.2} - \frac{3.2}{x^{2}}) + 6.4$

Step 3.6

Simplify each term.

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Step 3.6.1

Factor $2$ out of $2 x^{2}$ .

$\frac{2 (x^{2})}{3.2} + (x^{2} + 1) (\frac{1}{3.2} - \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.2

Factor $3.2$ out of $3.2$ .

$\frac{2 (x^{2})}{3.2 (1)} + (x^{2} + 1) (\frac{1}{3.2} - \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.3

Separate fractions.

$\frac{2}{3.2} \cdot \frac{x^{2}}{1} + (x^{2} + 1) (\frac{1}{3.2} - \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.4

Divide $2$ by $3.2$ .

$0.625 \frac{x^{2}}{1} + (x^{2} + 1) (\frac{1}{3.2} - \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.5

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

$0.625 x^{2} + (x^{2} + 1) (\frac{1}{3.2} - \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.6

Divide $1$ by $3.2$ .

$0.625 x^{2} + (x^{2} + 1) (0.3125 - \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.7

Expand $(x^{2} + 1) (0.3125 - \frac{3.2}{x^{2}})$ using the FOIL Method.

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Step 3.6.7.1

Apply the distributive property.

$0.625 x^{2} + x^{2} (0.3125 - \frac{3.2}{x^{2}}) + 1 (0.3125 - \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.7.2

Apply the distributive property.

$0.625 x^{2} + x^{2} \cdot 0.3125 + x^{2} (- \frac{3.2}{x^{2}}) + 1 (0.3125 - \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.7.3

Apply the distributive property.

$0.625 x^{2} + x^{2} \cdot 0.3125 + x^{2} (- \frac{3.2}{x^{2}}) + 1 \cdot 0.3125 + 1 (- \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.8

Simplify and combine like terms.

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Step 3.6.8.1

Simplify each term.

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Step 3.6.8.1.1

Move $0.3125$ to the left of $x^{2}$ .

$0.625 x^{2} + 0.3125 \cdot x^{2} + x^{2} (- \frac{3.2}{x^{2}}) + 1 \cdot 0.3125 + 1 (- \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.8.1.2

Rewrite using the commutative property of multiplication.

$0.625 x^{2} + 0.3125 x^{2} - x^{2} \frac{3.2}{x^{2}} + 1 \cdot 0.3125 + 1 (- \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.8.1.3

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

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Step 3.6.8.1.3.1

Factor $x^{2}$ out of $- x^{2}$ .

$0.625 x^{2} + 0.3125 x^{2} + x^{2} \cdot - 1 \frac{3.2}{x^{2}} + 1 \cdot 0.3125 + 1 (- \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.8.1.3.2

Cancel the common factor.

$0.625 x^{2} + 0.3125 x^{2} + x^{2} \cdot - 1 \frac{3.2}{x^{2}} + 1 \cdot 0.3125 + 1 (- \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.8.1.3.3

Rewrite the expression.

$0.625 x^{2} + 0.3125 x^{2} - 1 \cdot 3.2 + 1 \cdot 0.3125 + 1 (- \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.8.1.4

Multiply $- 1$ by $3.2$ .

$0.625 x^{2} + 0.3125 x^{2} - 3.2 + 1 \cdot 0.3125 + 1 (- \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.8.1.5

Multiply $0.3125$ by $1$ .

$0.625 x^{2} + 0.3125 x^{2} - 3.2 + 0.3125 + 1 (- \frac{3.2}{x^{2}}) + 6.4$

Step 3.6.8.1.6

Multiply $- \frac{3.2}{x^{2}}$ by $1$ .

$0.625 x^{2} + 0.3125 x^{2} - 3.2 + 0.3125 - \frac{3.2}{x^{2}} + 6.4$

Step 3.6.8.2

Add $- 3.2$ and $0.3125$ .

$0.625 x^{2} + 0.3125 x^{2} - 2.8875 - \frac{3.2}{x^{2}} + 6.4$

Step 3.7

Add $0.625 x^{2}$ and $0.3125 x^{2}$ .

$0.9375 x^{2} - 2.8875 - \frac{3.2}{x^{2}} + 6.4$

Step 3.8

Add $- 2.8875$ and $6.4$ .

$0.9375 x^{2} - \frac{3.2}{x^{2}} + 3.5125$