Calculus Examples

$y = (4 x^{2} + 9) (2 x - 3 + \frac{7}{x})$

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 $f (x) = 4 x^{2} + 9$ and $g (x) = 2 x - 3 + \frac{7}{x}$ .

$(4 x^{2} + 9) \frac{d}{d x} [2 x - 3 + \frac{7}{x}] + (2 x - 3 + \frac{7}{x}) \frac{d}{d x} [4 x^{2} + 9]$

Step 2

By the Sum Rule, the derivative of $2 x - 3 + \frac{7}{x}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 3] + \frac{d}{d x} [\frac{7}{x}]$ .

$(4 x^{2} + 9) (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 3] + \frac{d}{d x} [\frac{7}{x}]) + (2 x - 3 + \frac{7}{x}) \frac{d}{d x} [4 x^{2} + 9]$

Step 3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$(4 x^{2} + 9) (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 3] + \frac{d}{d x} [\frac{7}{x}]) + (2 x - 3 + \frac{7}{x}) \frac{d}{d x} [4 x^{2} + 9]$

Step 3.2

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

$(4 x^{2} + 9) (2 \cdot 1 + \frac{d}{d x} [- 3] + \frac{d}{d x} [\frac{7}{x}]) + (2 x - 3 + \frac{7}{x}) \frac{d}{d x} [4 x^{2} + 9]$

Step 3.3

Multiply $2$ by $1$ .

$(4 x^{2} + 9) (2 + \frac{d}{d x} [- 3] + \frac{d}{d x} [\frac{7}{x}]) + (2 x - 3 + \frac{7}{x}) \frac{d}{d x} [4 x^{2} + 9]$

Step 4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$(4 x^{2} + 9) (2 + 0 + \frac{d}{d x} [\frac{7}{x}]) + (2 x - 3 + \frac{7}{x}) \frac{d}{d x} [4 x^{2} + 9]$

Step 5

Evaluate $\frac{d}{d x} [\frac{7}{x}]$ .

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Step 5.1

Since $7$ is constant with respect to $x$ , the derivative of $\frac{7}{x}$ with respect to $x$ is $7 \frac{d}{d x} [\frac{1}{x}]$ .

$(4 x^{2} + 9) (2 + 0 + 7 \frac{d}{d x} [\frac{1}{x}]) + (2 x - 3 + \frac{7}{x}) \frac{d}{d x} [4 x^{2} + 9]$

Step 5.2

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

$(4 x^{2} + 9) (2 + 0 + 7 \frac{d}{d x} [x^{- 1}]) + (2 x - 3 + \frac{7}{x}) \frac{d}{d x} [4 x^{2} + 9]$

Step 5.3

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

$(4 x^{2} + 9) (2 + 0 + 7 (- x^{- 2})) + (2 x - 3 + \frac{7}{x}) \frac{d}{d x} [4 x^{2} + 9]$

Step 5.4

Multiply $- 1$ by $7$ .

$(4 x^{2} + 9) (2 + 0 - 7 x^{- 2}) + (2 x - 3 + \frac{7}{x}) \frac{d}{d x} [4 x^{2} + 9]$

Step 6

Simplify.

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Step 6.1

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

$(4 x^{2} + 9) (2 + 0 - 7 \frac{1}{x^{2}}) + (2 x - 3 + \frac{7}{x}) \frac{d}{d x} [4 x^{2} + 9]$

Step 6.2

Combine terms.

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Step 6.2.1

Add $2$ and $0$ .

$(4 x^{2} + 9) (2 - 7 \frac{1}{x^{2}}) + (2 x - 3 + \frac{7}{x}) \frac{d}{d x} [4 x^{2} + 9]$

Step 6.2.2

Combine $- 7$ and $\frac{1}{x^{2}}$ .

$(4 x^{2} + 9) (2 + \frac{- 7}{x^{2}}) + (2 x - 3 + \frac{7}{x}) \frac{d}{d x} [4 x^{2} + 9]$

Step 6.2.3

Move the negative in front of the fraction.

$(4 x^{2} + 9) (2 - \frac{7}{x^{2}}) + (2 x - 3 + \frac{7}{x}) \frac{d}{d x} [4 x^{2} + 9]$

Step 6.3

Reorder terms.

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + (2 x - 3 + \frac{7}{x}) \frac{d}{d x} [4 x^{2} + 9]$

Step 7

By the Sum Rule, the derivative of $4 x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [9]$ .

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + (2 x - 3 + \frac{7}{x}) (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [9])$

Step 8

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 8.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + (2 x - 3 + \frac{7}{x}) (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [9])$

Step 8.2

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

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + (2 x - 3 + \frac{7}{x}) (4 (2 x) + \frac{d}{d x} [9])$

Step 8.3

Multiply $2$ by $4$ .

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + (2 x - 3 + \frac{7}{x}) (8 x + \frac{d}{d x} [9])$

Step 9

Differentiate using the Constant Rule.

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Step 9.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + (2 x - 3 + \frac{7}{x}) (8 x + 0)$

Step 9.2

Add $8 x$ and $0$ .

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + (2 x - 3 + \frac{7}{x}) (8 x)$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + 2 x (8 x) - 3 (8 x) + \frac{7}{x} (8 x)$

Step 10.2

Combine terms.

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Step 10.2.1

Multiply $8$ by $2$ .

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + 16 x \cdot x - 3 (8 x) + \frac{7}{x} (8 x)$

Step 10.2.2

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

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + 16 (x^{1} x) - 3 (8 x) + \frac{7}{x} (8 x)$

Step 10.2.3

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

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + 16 (x^{1} x^{1}) - 3 (8 x) + \frac{7}{x} (8 x)$

Step 10.2.4

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

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + 16 x^{1 + 1} - 3 (8 x) + \frac{7}{x} (8 x)$

Step 10.2.5

Add $1$ and $1$ .

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + 16 x^{2} - 3 (8 x) + \frac{7}{x} (8 x)$

Step 10.2.6

Multiply $8$ by $- 3$ .

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + 16 x^{2} - 24 x + \frac{7}{x} (8 x)$

Step 10.2.7

Combine $8$ and $\frac{7}{x}$ .

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + 16 x^{2} - 24 x + \frac{8 \cdot 7}{x} x$

Step 10.2.8

Multiply $8$ by $7$ .

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + 16 x^{2} - 24 x + \frac{56}{x} x$

Step 10.2.9

Combine $\frac{56}{x}$ and $x$ .

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + 16 x^{2} - 24 x + \frac{56 x}{x}$

Step 10.2.10

Cancel the common factor of $x$ .

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Step 10.2.10.1

Cancel the common factor.

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + 16 x^{2} - 24 x + \frac{56 x}{x}$

Step 10.2.10.2

Divide $56$ by $1$ .

$(4 x^{2} + 9) (- \frac{7}{x^{2}} + 2) + 16 x^{2} - 24 x + 56$

Step 10.3

Reorder terms.

$16 x^{2} + (- \frac{7}{x^{2}} + 2) (4 x^{2} + 9) - 24 x + 56$

Step 10.4

Simplify each term.

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Step 10.4.1

Expand $(- \frac{7}{x^{2}} + 2) (4 x^{2} + 9)$ using the FOIL Method.

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Step 10.4.1.1

Apply the distributive property.

$16 x^{2} - \frac{7}{x^{2}} (4 x^{2} + 9) + 2 (4 x^{2} + 9) - 24 x + 56$

Step 10.4.1.2

Apply the distributive property.

$16 x^{2} - \frac{7}{x^{2}} (4 x^{2}) - \frac{7}{x^{2}} \cdot 9 + 2 (4 x^{2} + 9) - 24 x + 56$

Step 10.4.1.3

Apply the distributive property.

$16 x^{2} - \frac{7}{x^{2}} (4 x^{2}) - \frac{7}{x^{2}} \cdot 9 + 2 (4 x^{2}) + 2 \cdot 9 - 24 x + 56$

Step 10.4.2

Simplify and combine like terms.

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Step 10.4.2.1

Simplify each term.

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Step 10.4.2.1.1

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

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Step 10.4.2.1.1.1

Move the leading negative in $- \frac{7}{x^{2}}$ into the numerator.

$16 x^{2} + \frac{- 7}{x^{2}} (4 x^{2}) - \frac{7}{x^{2}} \cdot 9 + 2 (4 x^{2}) + 2 \cdot 9 - 24 x + 56$

Step 10.4.2.1.1.2

Factor $x^{2}$ out of $4 x^{2}$ .

$16 x^{2} + \frac{- 7}{x^{2}} (x^{2} \cdot 4) - \frac{7}{x^{2}} \cdot 9 + 2 (4 x^{2}) + 2 \cdot 9 - 24 x + 56$

Step 10.4.2.1.1.3

Cancel the common factor.

$16 x^{2} + \frac{- 7}{x^{2}} (x^{2} \cdot 4) - \frac{7}{x^{2}} \cdot 9 + 2 (4 x^{2}) + 2 \cdot 9 - 24 x + 56$

Step 10.4.2.1.1.4

Rewrite the expression.

$16 x^{2} - 7 \cdot 4 - \frac{7}{x^{2}} \cdot 9 + 2 (4 x^{2}) + 2 \cdot 9 - 24 x + 56$

Step 10.4.2.1.2

Multiply $- 7$ by $4$ .

$16 x^{2} - 28 - \frac{7}{x^{2}} \cdot 9 + 2 (4 x^{2}) + 2 \cdot 9 - 24 x + 56$

Step 10.4.2.1.3

Multiply $- \frac{7}{x^{2}} \cdot 9$ .

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Step 10.4.2.1.3.1

Multiply $9$ by $- 1$ .

$16 x^{2} - 28 - 9 \frac{7}{x^{2}} + 2 (4 x^{2}) + 2 \cdot 9 - 24 x + 56$

Step 10.4.2.1.3.2

Combine $- 9$ and $\frac{7}{x^{2}}$ .

$16 x^{2} - 28 + \frac{- 9 \cdot 7}{x^{2}} + 2 (4 x^{2}) + 2 \cdot 9 - 24 x + 56$

Step 10.4.2.1.3.3

Multiply $- 9$ by $7$ .

$16 x^{2} - 28 + \frac{- 63}{x^{2}} + 2 (4 x^{2}) + 2 \cdot 9 - 24 x + 56$

Step 10.4.2.1.4

Move the negative in front of the fraction.

$16 x^{2} - 28 - \frac{63}{x^{2}} + 2 (4 x^{2}) + 2 \cdot 9 - 24 x + 56$

Step 10.4.2.1.5

Multiply $4$ by $2$ .

$16 x^{2} - 28 - \frac{63}{x^{2}} + 8 x^{2} + 2 \cdot 9 - 24 x + 56$

Step 10.4.2.1.6

Multiply $2$ by $9$ .

$16 x^{2} - 28 - \frac{63}{x^{2}} + 8 x^{2} + 18 - 24 x + 56$

Step 10.4.2.2

Add $- 28$ and $18$ .

$16 x^{2} - \frac{63}{x^{2}} + 8 x^{2} - 10 - 24 x + 56$

Step 10.5

Add $16 x^{2}$ and $8 x^{2}$ .

$24 x^{2} - \frac{63}{x^{2}} - 10 - 24 x + 56$

Step 10.6

Add $- 10$ and $56$ .

$24 x^{2} - \frac{63}{x^{2}} - 24 x + 46$