Calculus Examples

$(x + 8) (\frac{24}{x} + 12)$

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 $f (x) = x + 8$ and $g (x) = \frac{24}{x} + 12$ .

$(x + 8) \frac{d}{d x} [\frac{24}{x} + 12] + (\frac{24}{x} + 12) \frac{d}{d x} [x + 8]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $\frac{24}{x} + 12$ with respect to $x$ is $\frac{d}{d x} [\frac{24}{x}] + \frac{d}{d x} [12]$ .

$(x + 8) (\frac{d}{d x} [\frac{24}{x}] + \frac{d}{d x} [12]) + (\frac{24}{x} + 12) \frac{d}{d x} [x + 8]$

Step 2.2

Since $24$ is constant with respect to $x$ , the derivative of $\frac{24}{x}$ with respect to $x$ is $24 \frac{d}{d x} [\frac{1}{x}]$ .

$(x + 8) (24 \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [12]) + (\frac{24}{x} + 12) \frac{d}{d x} [x + 8]$

Step 2.3

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

$(x + 8) (24 \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [12]) + (\frac{24}{x} + 12) \frac{d}{d x} [x + 8]$

Step 2.4

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

$(x + 8) (24 (- x^{- 2}) + \frac{d}{d x} [12]) + (\frac{24}{x} + 12) \frac{d}{d x} [x + 8]$

Step 2.5

Multiply $- 1$ by $24$ .

$(x + 8) (- 24 x^{- 2} + \frac{d}{d x} [12]) + (\frac{24}{x} + 12) \frac{d}{d x} [x + 8]$

Step 2.6

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$(x + 8) (- 24 x^{- 2} + 0) + (\frac{24}{x} + 12) \frac{d}{d x} [x + 8]$

Step 2.7

Add $- 24 x^{- 2}$ and $0$ .

$(x + 8) (- 24 x^{- 2}) + (\frac{24}{x} + 12) \frac{d}{d x} [x + 8]$

Step 2.8

By the Sum Rule, the derivative of $x + 8$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [8]$ .

$(x + 8) (- 24 x^{- 2}) + (\frac{24}{x} + 12) (\frac{d}{d x} [x] + \frac{d}{d x} [8])$

Step 2.9

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

$(x + 8) (- 24 x^{- 2}) + (\frac{24}{x} + 12) (1 + \frac{d}{d x} [8])$

Step 2.10

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$(x + 8) (- 24 x^{- 2}) + (\frac{24}{x} + 12) (1 + 0)$

Step 2.11

Simplify the expression.

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Step 2.11.1

Add $1$ and $0$ .

$(x + 8) (- 24 x^{- 2}) + (\frac{24}{x} + 12) \cdot 1$

Step 2.11.2

Multiply $\frac{24}{x} + 12$ by $1$ .

$(x + 8) (- 24 x^{- 2}) + \frac{24}{x} + 12$

Step 3

To write $(x + 8) (- 24 x^{- 2})$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{(x + 8) (- 24 x^{- 2}) x}{x} + \frac{24}{x} + 12$

Step 4

Combine the numerators over the common denominator.

$\frac{(x + 8) (- 24 x^{- 2}) x + 24}{x} + 12$

Step 5

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

$\frac{(x + 8) (- 24 (x^{1} x^{- 2})) + 24}{x} + 12$

Step 6

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

$\frac{(x + 8) (- 24 x^{1 - 2}) + 24}{x} + 12$

Step 7

Subtract $2$ from $1$ .

$\frac{(x + 8) (- 24 x^{- 1}) + 24}{x} + 12$

Step 8

Simplify.

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Step 8.1

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

$\frac{(x + 8) (- 24 \frac{1}{x}) + 24}{x} + 12$

Step 8.2

Apply the distributive property.

$\frac{x (- 24 \frac{1}{x}) + 8 (- 24 \frac{1}{x}) + 24}{x} + 12$

Step 8.3

Combine terms.

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Step 8.3.1

Combine $- 24$ and $\frac{1}{x}$ .

$\frac{x \frac{- 24}{x} + 8 (- 24 \frac{1}{x}) + 24}{x} + 12$

Step 8.3.2

Move the negative in front of the fraction.

$\frac{x (- \frac{24}{x}) + 8 (- 24 \frac{1}{x}) + 24}{x} + 12$

Step 8.3.3

Combine $x$ and $\frac{24}{x}$ .

$\frac{- \frac{x \cdot 24}{x} + 8 (- 24 \frac{1}{x}) + 24}{x} + 12$

Step 8.3.4

Move $24$ to the left of $x$ .

$\frac{- \frac{24 \cdot x}{x} + 8 (- 24 \frac{1}{x}) + 24}{x} + 12$

Step 8.3.5

Cancel the common factor of $x$ .

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Step 8.3.5.1

Cancel the common factor.

$\frac{- \frac{24 x}{x} + 8 (- 24 \frac{1}{x}) + 24}{x} + 12$

Step 8.3.5.2

Divide $24$ by $1$ .

$\frac{- 1 \cdot 24 + 8 (- 24 \frac{1}{x}) + 24}{x} + 12$

Step 8.3.6

Multiply $- 1$ by $24$ .

$\frac{- 24 + 8 (- 24 \frac{1}{x}) + 24}{x} + 12$

Step 8.3.7

Combine $- 24$ and $\frac{1}{x}$ .

$\frac{- 24 + 8 \frac{- 24}{x} + 24}{x} + 12$

Step 8.3.8

Move the negative in front of the fraction.

$\frac{- 24 + 8 (- \frac{24}{x}) + 24}{x} + 12$

Step 8.3.9

Multiply $- 1$ by $8$ .

$\frac{- 24 - 8 \frac{24}{x} + 24}{x} + 12$

Step 8.3.10

Combine $- 8$ and $\frac{24}{x}$ .

$\frac{- 24 + \frac{- 8 \cdot 24}{x} + 24}{x} + 12$

Step 8.3.11

Multiply $- 8$ by $24$ .

$\frac{- 24 + \frac{- 192}{x} + 24}{x} + 12$

Step 8.3.12

Move the negative in front of the fraction.

$\frac{- 24 - \frac{192}{x} + 24}{x} + 12$

Step 8.3.13

Add $- 24$ and $24$ .

$\frac{- \frac{192}{x} + 0}{x} + 12$

Step 8.3.14

Add $- \frac{192}{x}$ and $0$ .

$\frac{- \frac{192}{x}}{x} + 12$

Step 8.3.15

Rewrite $\frac{- \frac{192}{x}}{x}$ as a product.

$- \frac{192}{x} \cdot \frac{1}{x} + 12$

Step 8.3.16

Multiply $\frac{1}{x}$ by $\frac{192}{x}$ .

$- \frac{192}{x \cdot x} + 12$

Step 8.3.17

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

$- \frac{192}{x^{1} x} + 12$

Step 8.3.18

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

$- \frac{192}{x^{1} x^{1}} + 12$

Step 8.3.19

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

$- \frac{192}{x^{1 + 1}} + 12$

Step 8.3.20

Add $1$ and $1$ .

$- \frac{192}{x^{2}} + 12$