Calculus Examples

$g (x) = \frac{2 x - 3}{7 x + 4}$ , $x = 4$

Step 1

Find the corresponding $y$ -value to $x = 4$ .

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Step 1.1

Substitute $4$ in for $x$ .

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

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

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

Step 1.2.2

Simplify $\frac{2 (4) - 3}{7 (4) + 4}$ .

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Step 1.2.2.1

Simplify the numerator.

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Step 1.2.2.1.1

Multiply $2$ by $4$ .

$y = \frac{8 - 3}{7 (4) + 4}$

Step 1.2.2.1.2

Subtract $3$ from $8$ .

$y = \frac{5}{7 (4) + 4}$

Step 1.2.2.2

Simplify the denominator.

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Step 1.2.2.2.1

Multiply $7$ by $4$ .

$y = \frac{5}{28 + 4}$

Step 1.2.2.2.2

Add $28$ and $4$ .

$y = \frac{5}{32}$

Step 2

Find the first derivative and evaluate at $x = 4$ and $y = \frac{5}{32}$ to find the slope of the tangent line.

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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 $f (x) = 2 x - 3$ and $g (x) = 7 x + 4$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $2$ by $1$ .

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

Step 2.2.5

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

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

Step 2.2.6

Simplify the expression.

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Step 2.2.6.1

Add $2$ and $0$ .

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

Step 2.2.6.2

Move $2$ to the left of $7 x + 4$ .

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

Step 2.2.7

By the Sum Rule, the derivative of $7 x + 4$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [4]$ .

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

Step 2.2.8

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

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

Step 2.2.9

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

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

Step 2.2.10

Multiply $7$ by $1$ .

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

Step 2.2.11

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 2.2.12

Simplify the expression.

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Step 2.2.12.1

Add $7$ and $0$ .

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

Step 2.2.12.2

Multiply $7$ by $- 1$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Apply the distributive property.

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

Step 2.3.3

Simplify the numerator.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Multiply $7$ by $2$ .

$\frac{14 x + 2 \cdot 4 - 7 (2 x) - 7 \cdot - 3}{{(7 x + 4)}^{2}}$

Step 2.3.3.1.2

Multiply $2$ by $4$ .

$\frac{14 x + 8 - 7 (2 x) - 7 \cdot - 3}{{(7 x + 4)}^{2}}$

Step 2.3.3.1.3

Multiply $2$ by $- 7$ .

$\frac{14 x + 8 - 14 x - 7 \cdot - 3}{{(7 x + 4)}^{2}}$

Step 2.3.3.1.4

Multiply $- 7$ by $- 3$ .

$\frac{14 x + 8 - 14 x + 21}{{(7 x + 4)}^{2}}$

Step 2.3.3.2

Combine the opposite terms in $14 x + 8 - 14 x + 21$ .

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Step 2.3.3.2.1

Subtract $14 x$ from $14 x$ .

$\frac{0 + 8 + 21}{{(7 x + 4)}^{2}}$

Step 2.3.3.2.2

Add $0$ and $8$ .

$\frac{8 + 21}{{(7 x + 4)}^{2}}$

Step 2.3.3.3

Add $8$ and $21$ .

$\frac{29}{{(7 x + 4)}^{2}}$

Step 2.4

Evaluate the derivative at $x = 4$ .

$\frac{29}{{(7 (4) + 4)}^{2}}$

Step 2.5

Simplify the denominator.

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Step 2.5.1

Multiply $7$ by $4$ .

$\frac{29}{{(28 + 4)}^{2}}$

Step 2.5.2

Add $28$ and $4$ .

$\frac{29}{32^{2}}$

Step 2.5.3

Raise $32$ to the power of $2$ .

$\frac{29}{1024}$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $\frac{29}{1024}$ and a given point $(4, \frac{5}{32})$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{5}{32}) = \frac{29}{1024} \cdot (x - (4))$

Step 3.2

Simplify the equation and keep it in point-slope form.

$y - \frac{5}{32} = \frac{29}{1024} \cdot (x - 4)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $\frac{29}{1024} \cdot (x - 4)$ .

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Step 3.3.1.1

Rewrite.

$y - \frac{5}{32} = 0 + 0 + \frac{29}{1024} \cdot (x - 4)$

Step 3.3.1.2

Simplify by adding zeros.

$y - \frac{5}{32} = \frac{29}{1024} \cdot (x - 4)$

Step 3.3.1.3

Apply the distributive property.

$y - \frac{5}{32} = \frac{29}{1024} x + \frac{29}{1024} \cdot - 4$

Step 3.3.1.4

Combine $\frac{29}{1024}$ and $x$ .

$y - \frac{5}{32} = \frac{29 x}{1024} + \frac{29}{1024} \cdot - 4$

Step 3.3.1.5

Cancel the common factor of $4$ .

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Step 3.3.1.5.1

Factor $4$ out of $1024$ .

$y - \frac{5}{32} = \frac{29 x}{1024} + \frac{29}{4 (256)} \cdot - 4$

Step 3.3.1.5.2

Factor $4$ out of $- 4$ .

$y - \frac{5}{32} = \frac{29 x}{1024} + \frac{29}{4 \cdot 256} \cdot (4 \cdot - 1)$

Step 3.3.1.5.3

Cancel the common factor.

$y - \frac{5}{32} = \frac{29 x}{1024} + \frac{29}{4 \cdot 256} \cdot (4 \cdot - 1)$

Step 3.3.1.5.4

Rewrite the expression.

$y - \frac{5}{32} = \frac{29 x}{1024} + \frac{29}{256} \cdot - 1$

Step 3.3.1.6

Combine $\frac{29}{256}$ and $- 1$ .

$y - \frac{5}{32} = \frac{29 x}{1024} + \frac{29 \cdot - 1}{256}$

Step 3.3.1.7

Simplify the expression.

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Step 3.3.1.7.1

Multiply $29$ by $- 1$ .

$y - \frac{5}{32} = \frac{29 x}{1024} + \frac{- 29}{256}$

Step 3.3.1.7.2

Move the negative in front of the fraction.

$y - \frac{5}{32} = \frac{29 x}{1024} - \frac{29}{256}$

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.2.1

Add $\frac{5}{32}$ to both sides of the equation.

$y = \frac{29 x}{1024} - \frac{29}{256} + \frac{5}{32}$

Step 3.3.2.2

To write $\frac{5}{32}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = \frac{29 x}{1024} - \frac{29}{256} + \frac{5}{32} \cdot \frac{8}{8}$

Step 3.3.2.3

Write each expression with a common denominator of $256$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.2.3.1

Multiply $\frac{5}{32}$ by $\frac{8}{8}$ .

$y = \frac{29 x}{1024} - \frac{29}{256} + \frac{5 \cdot 8}{32 \cdot 8}$

Step 3.3.2.3.2

Multiply $32$ by $8$ .

$y = \frac{29 x}{1024} - \frac{29}{256} + \frac{5 \cdot 8}{256}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$y = \frac{29 x}{1024} + \frac{- 29 + 5 \cdot 8}{256}$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $5$ by $8$ .

$y = \frac{29 x}{1024} + \frac{- 29 + 40}{256}$

Step 3.3.2.5.2

Add $- 29$ and $40$ .

$y = \frac{29 x}{1024} + \frac{11}{256}$

Step 3.3.3

Reorder terms.

$y = \frac{29}{1024} x + \frac{11}{256}$

Step 4