Calculus Examples

$2.7 (\frac{x}{2}) + (18 + 1.1 x) (\frac{120}{x})$

Step 1

By the Sum Rule, the derivative of $2.7 (\frac{x}{2}) + (18 + 1.1 x) (\frac{120}{x})$ with respect to $x$ is $\frac{d}{d x} [2.7 (\frac{x}{2})] + \frac{d}{d x} [(18 + 1.1 x) (\frac{120}{x})]$ .

$\frac{d}{d x} [2.7 (\frac{x}{2})] + \frac{d}{d x} [(18 + 1.1 x) (\frac{120}{x})]$

Step 2

Evaluate $\frac{d}{d x} [2.7 (\frac{x}{2})]$ .

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Step 2.1

Combine $2.7$ and $\frac{x}{2}$ .

$\frac{d}{d x} [\frac{2.7 x}{2}] + \frac{d}{d x} [(18 + 1.1 x) (\frac{120}{x})]$

Step 2.2

Since $\frac{2.7}{2}$ is constant with respect to $x$ , the derivative of $\frac{2.7 x}{2}$ with respect to $x$ is $\frac{2.7}{2} \frac{d}{d x} [x]$ .

$\frac{2.7}{2} \frac{d}{d x} [x] + \frac{d}{d x} [(18 + 1.1 x) (\frac{120}{x})]$

Step 2.3

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

$\frac{2.7}{2} \cdot 1 + \frac{d}{d x} [(18 + 1.1 x) (\frac{120}{x})]$

Step 2.4

Multiply $\frac{2.7}{2}$ by $1$ .

$\frac{2.7}{2} + \frac{d}{d x} [(18 + 1.1 x) (\frac{120}{x})]$

Step 3

Evaluate $\frac{d}{d x} [(18 + 1.1 x) (\frac{120}{x})]$ .

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Step 3.1

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 $f (x) = 18 + 1.1 x$ and $g (x) = \frac{120}{x}$ .

$\frac{2.7}{2} + (18 + 1.1 x) \frac{d}{d x} [\frac{120}{x}] + \frac{120}{x} \frac{d}{d x} [18 + 1.1 x]$

Step 3.2

Since $120$ is constant with respect to $x$ , the derivative of $\frac{120}{x}$ with respect to $x$ is $120 \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{2.7}{2} + (18 + 1.1 x) (120 \frac{d}{d x} [\frac{1}{x}]) + \frac{120}{x} \frac{d}{d x} [18 + 1.1 x]$

Step 3.3

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

$\frac{2.7}{2} + (18 + 1.1 x) (120 \frac{d}{d x} [x^{- 1}]) + \frac{120}{x} \frac{d}{d x} [18 + 1.1 x]$

Step 3.4

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

$\frac{2.7}{2} + (18 + 1.1 x) (120 (- x^{- 2})) + \frac{120}{x} \frac{d}{d x} [18 + 1.1 x]$

Step 3.5

By the Sum Rule, the derivative of $18 + 1.1 x$ with respect to $x$ is $\frac{d}{d x} [18] + \frac{d}{d x} [1.1 x]$ .

$\frac{2.7}{2} + (18 + 1.1 x) (120 (- x^{- 2})) + \frac{120}{x} (\frac{d}{d x} [18] + \frac{d}{d x} [1.1 x])$

Step 3.6

Since $18$ is constant with respect to $x$ , the derivative of $18$ with respect to $x$ is $0$ .

$\frac{2.7}{2} + (18 + 1.1 x) (120 (- x^{- 2})) + \frac{120}{x} (0 + \frac{d}{d x} [1.1 x])$

Step 3.7

Since $1.1$ is constant with respect to $x$ , the derivative of $1.1 x$ with respect to $x$ is $1.1 \frac{d}{d x} [x]$ .

$\frac{2.7}{2} + (18 + 1.1 x) (120 (- x^{- 2})) + \frac{120}{x} (0 + 1.1 \frac{d}{d x} [x])$

Step 3.8

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

$\frac{2.7}{2} + (18 + 1.1 x) (120 (- x^{- 2})) + \frac{120}{x} (0 + 1.1 \cdot 1)$

Step 3.9

Multiply $- 1$ by $120$ .

$\frac{2.7}{2} + (18 + 1.1 x) (- 120 x^{- 2}) + \frac{120}{x} (0 + 1.1 \cdot 1)$

Step 3.10

Multiply $1.1$ by $1$ .

$\frac{2.7}{2} + (18 + 1.1 x) (- 120 x^{- 2}) + \frac{120}{x} (0 + 1.1)$

Step 3.11

Add $0$ and $1.1$ .

$\frac{2.7}{2} + (18 + 1.1 x) (- 120 x^{- 2}) + \frac{120}{x} \cdot 1.1$

Step 3.12

Combine $\frac{120}{x}$ and $1.1$ .

$\frac{2.7}{2} + (18 + 1.1 x) (- 120 x^{- 2}) + \frac{120 \cdot 1.1}{x}$

Step 3.13

Multiply $120$ by $1.1$ .

$\frac{2.7}{2} + (18 + 1.1 x) (- 120 x^{- 2}) + \frac{132}{x}$

Step 3.14

To write $(18 + 1.1 x) (- 120 x^{- 2})$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{2.7}{2} + \frac{(18 + 1.1 x) (- 120 x^{- 2}) x}{x} + \frac{132}{x}$

Step 3.15

Combine the numerators over the common denominator.

$\frac{2.7}{2} + \frac{(18 + 1.1 x) (- 120 x^{- 2}) x + 132}{x}$

Step 3.16

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

$\frac{2.7}{2} + \frac{(18 + 1.1 x) (- 120 (x^{1} x^{- 2})) + 132}{x}$

Step 3.17

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

$\frac{2.7}{2} + \frac{(18 + 1.1 x) (- 120 x^{1 - 2}) + 132}{x}$

Step 3.18

Subtract $2$ from $1$ .

$\frac{2.7}{2} + \frac{(18 + 1.1 x) (- 120 x^{- 1}) + 132}{x}$

Step 4

Simplify.

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Step 4.1

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

$\frac{2.7}{2} + \frac{(18 + 1.1 x) (- 120 \frac{1}{x}) + 132}{x}$

Step 4.2

Apply the distributive property.

$\frac{2.7}{2} + \frac{18 (- 120 \frac{1}{x}) + 1.1 x (- 120 \frac{1}{x}) + 132}{x}$

Step 4.3

Combine terms.

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Step 4.3.1

Combine $- 120$ and $\frac{1}{x}$ .

$\frac{2.7}{2} + \frac{18 \frac{- 120}{x} + 1.1 x (- 120 \frac{1}{x}) + 132}{x}$

Step 4.3.2

Move the negative in front of the fraction.

$\frac{2.7}{2} + \frac{18 (- \frac{120}{x}) + 1.1 x (- 120 \frac{1}{x}) + 132}{x}$

Step 4.3.3

Multiply $- 1$ by $18$ .

$\frac{2.7}{2} + \frac{- 18 \frac{120}{x} + 1.1 x (- 120 \frac{1}{x}) + 132}{x}$

Step 4.3.4

Combine $- 18$ and $\frac{120}{x}$ .

$\frac{2.7}{2} + \frac{\frac{- 18 \cdot 120}{x} + 1.1 x (- 120 \frac{1}{x}) + 132}{x}$

Step 4.3.5

Multiply $- 18$ by $120$ .

$\frac{2.7}{2} + \frac{\frac{- 2160}{x} + 1.1 x (- 120 \frac{1}{x}) + 132}{x}$

Step 4.3.6

Move the negative in front of the fraction.

$\frac{2.7}{2} + \frac{- \frac{2160}{x} + 1.1 x (- 120 \frac{1}{x}) + 132}{x}$

Step 4.3.7

Combine $- 120$ and $\frac{1}{x}$ .

$\frac{2.7}{2} + \frac{- \frac{2160}{x} + 1.1 x \frac{- 120}{x} + 132}{x}$

Step 4.3.8

Move the negative in front of the fraction.

$\frac{2.7}{2} + \frac{- \frac{2160}{x} + 1.1 x (- \frac{120}{x}) + 132}{x}$

Step 4.3.9

Multiply $- 1$ by $1.1$ .

$\frac{2.7}{2} + \frac{- \frac{2160}{x} - 1.1 x \frac{120}{x} + 132}{x}$

Step 4.3.10

Combine $- 1.1$ and $\frac{120}{x}$ .

$\frac{2.7}{2} + \frac{- \frac{2160}{x} + x \frac{- 1.1 \cdot 120}{x} + 132}{x}$

Step 4.3.11

Multiply $- 1.1$ by $120$ .

$\frac{2.7}{2} + \frac{- \frac{2160}{x} + x \frac{- 132}{x} + 132}{x}$

Step 4.3.12

Combine $x$ and $\frac{- 132}{x}$ .

$\frac{2.7}{2} + \frac{- \frac{2160}{x} + \frac{x \cdot - 132}{x} + 132}{x}$

Step 4.3.13

Move $- 132$ to the left of $x$ .

$\frac{2.7}{2} + \frac{- \frac{2160}{x} + \frac{- 132 \cdot x}{x} + 132}{x}$

Step 4.3.14

Cancel the common factor of $x$ .

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Step 4.3.14.1

Cancel the common factor.

$\frac{2.7}{2} + \frac{- \frac{2160}{x} + \frac{- 132 x}{x} + 132}{x}$

Step 4.3.14.2

Divide $- 132$ by $1$ .

$\frac{2.7}{2} + \frac{- \frac{2160}{x} - 132 + 132}{x}$

Step 4.3.15

Add $- 132$ and $132$ .

$\frac{2.7}{2} + \frac{- \frac{2160}{x} + 0}{x}$

Step 4.3.16

Add $- \frac{2160}{x}$ and $0$ .

$\frac{2.7}{2} + \frac{- \frac{2160}{x}}{x}$

Step 4.3.17

Rewrite $\frac{- \frac{2160}{x}}{x}$ as a product.

$\frac{2.7}{2} - \frac{2160}{x} \cdot \frac{1}{x}$

Step 4.3.18

Multiply $\frac{1}{x}$ by $\frac{2160}{x}$ .

$\frac{2.7}{2} - \frac{2160}{x \cdot x}$

Step 4.3.19

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

$\frac{2.7}{2} - \frac{2160}{x^{1} x}$

Step 4.3.20

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

$\frac{2.7}{2} - \frac{2160}{x^{1} x^{1}}$

Step 4.3.21

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

$\frac{2.7}{2} - \frac{2160}{x^{1 + 1}}$

Step 4.3.22

Add $1$ and $1$ .

$\frac{2.7}{2} - \frac{2160}{x^{2}}$

Step 4.4

Reorder terms.

$- \frac{2160}{x^{2}} + \frac{2.7}{2}$

Step 4.5

Divide $2.7$ by $2$ .

$- \frac{2160}{x^{2}} + 1.35$