Linear Algebra Examples

$\frac{x + 7}{x} + \frac{6}{3 x + 21}$

Step 1

By the Sum Rule, the derivative of $\frac{x + 7}{x} + \frac{6}{3 x + 21}$ with respect to $x$ is $\frac{d}{d x} [\frac{x + 7}{x}] + \frac{d}{d x} [\frac{6}{3 x + 21}]$ .

$\frac{d}{d x} [\frac{x + 7}{x}] + \frac{d}{d x} [\frac{6}{3 x + 21}]$

Step 2

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

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Step 2.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x + 7$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [x + 7] - (x + 7) \frac{d}{d x} [x]}{x^{2}} + \frac{d}{d x} [\frac{6}{3 x + 21}]$

Step 2.2

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

$\frac{x (\frac{d}{d x} [x] + \frac{d}{d x} [7]) - (x + 7) \frac{d}{d x} [x]}{x^{2}} + \frac{d}{d x} [\frac{6}{3 x + 21}]$

Step 2.3

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

$\frac{x (1 + \frac{d}{d x} [7]) - (x + 7) \frac{d}{d x} [x]}{x^{2}} + \frac{d}{d x} [\frac{6}{3 x + 21}]$

Step 2.4

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{x (1 + 0) - (x + 7) \frac{d}{d x} [x]}{x^{2}} + \frac{d}{d x} [\frac{6}{3 x + 21}]$

Step 2.5

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

$\frac{x (1 + 0) - (x + 7) \cdot 1}{x^{2}} + \frac{d}{d x} [\frac{6}{3 x + 21}]$

Step 2.6

Add $1$ and $0$ .

$\frac{x \cdot 1 - (x + 7) \cdot 1}{x^{2}} + \frac{d}{d x} [\frac{6}{3 x + 21}]$

Step 2.7

Multiply $x$ by $1$ .

$\frac{x - (x + 7) \cdot 1}{x^{2}} + \frac{d}{d x} [\frac{6}{3 x + 21}]$

Step 2.8

Multiply $- 1$ by $1$ .

$\frac{x - (x + 7)}{x^{2}} + \frac{d}{d x} [\frac{6}{3 x + 21}]$

Step 3

Evaluate $\frac{d}{d x} [\frac{6}{3 x + 21}]$ .

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Step 3.1

Cancel the common factor of $6$ and $3 x + 21$ .

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Step 3.1.1

Factor $3$ out of $6$ .

$\frac{x - (x + 7)}{x^{2}} + \frac{d}{d x} [\frac{3 \cdot 2}{3 x + 21}]$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $3$ out of $3 x$ .

$\frac{x - (x + 7)}{x^{2}} + \frac{d}{d x} [\frac{3 \cdot 2}{3 (x) + 21}]$

Step 3.1.2.2

Factor $3$ out of $21$ .

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

Step 3.1.2.3

Factor $3$ out of $3 (x) + 3 (7)$ .

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

Step 3.1.2.4

Cancel the common factor.

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

Step 3.1.2.5

Rewrite the expression.

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

Step 3.2

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x + 7}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x + 7}]$ .

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

Step 3.3

Rewrite $\frac{1}{x + 7}$ as ${(x + 7)}^{- 1}$ .

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

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x + 7$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $x + 7$ .

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

Step 3.4.2

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

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

Step 3.4.3

Replace all occurrences of $u$ with $x + 7$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{x - (x + 7)}{x^{2}} + 2 (- {(x + 7)}^{- 2} (1 + 0))$

Step 3.8

Add $1$ and $0$ .

$\frac{x - (x + 7)}{x^{2}} + 2 (- {(x + 7)}^{- 2} \cdot 1)$

Step 3.9

Multiply $- 1$ by $1$ .

$\frac{x - (x + 7)}{x^{2}} + 2 (- {(x + 7)}^{- 2})$

Step 3.10

Multiply $- 1$ by $2$ .

$\frac{x - (x + 7)}{x^{2}} - 2 {(x + 7)}^{- 2}$

Step 4

Simplify.

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Step 4.1

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

$\frac{x - (x + 7)}{x^{2}} - 2 \frac{1}{{(x + 7)}^{2}}$

Step 4.2

Apply the distributive property.

$\frac{x - x - 1 \cdot 7}{x^{2}} - 2 \frac{1}{{(x + 7)}^{2}}$

Step 4.3

Combine terms.

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Step 4.3.1

Multiply $- 1$ by $7$ .

$\frac{x - x - 7}{x^{2}} - 2 \frac{1}{{(x + 7)}^{2}}$

Step 4.3.2

Subtract $x$ from $x$ .

$\frac{0 - 7}{x^{2}} - 2 \frac{1}{{(x + 7)}^{2}}$

Step 4.3.3

Subtract $7$ from $0$ .

$\frac{- 7}{x^{2}} - 2 \frac{1}{{(x + 7)}^{2}}$

Step 4.3.4

Move the negative in front of the fraction.

$- \frac{7}{x^{2}} - 2 \frac{1}{{(x + 7)}^{2}}$

Step 4.3.5

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

$- \frac{7}{x^{2}} + \frac{- 2}{{(x + 7)}^{2}}$

Step 4.3.6

Move the negative in front of the fraction.

$- \frac{7}{x^{2}} - \frac{2}{{(x + 7)}^{2}}$