Calculus Examples

$f (x) = \frac{3 x - 4}{4 x - 3}$ , $x = 1$

Step 1

Find the corresponding $y$ -value to $x = 1$ .

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Step 1.1

Substitute $1$ in for $x$ .

$y = \frac{3 (1) - 4}{4 (1) - 3}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = \frac{3 (1) - 4}{4 (1) - 3}$

Step 1.2.2

Simplify $\frac{3 (1) - 4}{4 (1) - 3}$ .

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Step 1.2.2.1

Simplify the numerator.

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Step 1.2.2.1.1

Multiply $3$ by $1$ .

$y = \frac{3 - 4}{4 (1) - 3}$

Step 1.2.2.1.2

Subtract $4$ from $3$ .

$y = \frac{- 1}{4 (1) - 3}$

Step 1.2.2.2

Simplify the denominator.

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Step 1.2.2.2.1

Multiply $4$ by $1$ .

$y = \frac{- 1}{4 - 3}$

Step 1.2.2.2.2

Subtract $3$ from $4$ .

$y = \frac{- 1}{1}$

Step 1.2.2.3

Divide $- 1$ by $1$ .

$y = - 1$

Step 2

Find the first derivative and evaluate at $x = 1$ and $y = - 1$ 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 $f (x) = 3 x - 4$ and $g (x) = 4 x - 3$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $3 x - 4$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 4]$ .

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

Step 2.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $3$ by $1$ .

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

Step 2.2.5

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

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

Step 2.2.6

Simplify the expression.

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Step 2.2.6.1

Add $3$ and $0$ .

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

Step 2.2.6.2

Move $3$ to the left of $4 x - 3$ .

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

Step 2.2.7

By the Sum Rule, the derivative of $4 x - 3$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]$ .

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

Step 2.2.8

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

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

Step 2.2.9

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

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

Step 2.2.10

Multiply $4$ by $1$ .

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

Step 2.2.11

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

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

Step 2.2.12

Simplify the expression.

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Step 2.2.12.1

Add $4$ and $0$ .

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

Step 2.2.12.2

Multiply $4$ by $- 1$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Apply the distributive property.

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

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 $4$ by $3$ .

$\frac{12 x + 3 \cdot - 3 - 4 (3 x) - 4 \cdot - 4}{{(4 x - 3)}^{2}}$

Step 2.3.3.1.2

Multiply $3$ by $- 3$ .

$\frac{12 x - 9 - 4 (3 x) - 4 \cdot - 4}{{(4 x - 3)}^{2}}$

Step 2.3.3.1.3

Multiply $3$ by $- 4$ .

$\frac{12 x - 9 - 12 x - 4 \cdot - 4}{{(4 x - 3)}^{2}}$

Step 2.3.3.1.4

Multiply $- 4$ by $- 4$ .

$\frac{12 x - 9 - 12 x + 16}{{(4 x - 3)}^{2}}$

Step 2.3.3.2

Combine the opposite terms in $12 x - 9 - 12 x + 16$ .

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Step 2.3.3.2.1

Subtract $12 x$ from $12 x$ .

$\frac{0 - 9 + 16}{{(4 x - 3)}^{2}}$

Step 2.3.3.2.2

Subtract $9$ from $0$ .

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

Step 2.3.3.3

Add $- 9$ and $16$ .

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

Step 2.4

Evaluate the derivative at $x = 1$ .

$\frac{7}{{(4 (1) - 3)}^{2}}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the denominator.

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Step 2.5.1.1

Multiply $4$ by $1$ .

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

Step 2.5.1.2

Subtract $3$ from $4$ .

$\frac{7}{1^{2}}$

Step 2.5.1.3

One to any power is one.

$\frac{7}{1}$

Step 2.5.2

Divide $7$ by $1$ .

$7$

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 $7$ and a given point $(1, - 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 1) = 7 \cdot (x - (1))$

Step 3.2

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

$y + 1 = 7 \cdot (x - 1)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $7 \cdot (x - 1)$ .

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Step 3.3.1.1

Rewrite.

$y + 1 = 0 + 0 + 7 \cdot (x - 1)$

Step 3.3.1.2

Simplify by adding zeros.

$y + 1 = 7 \cdot (x - 1)$

Step 3.3.1.3

Apply the distributive property.

$y + 1 = 7 x + 7 \cdot - 1$

Step 3.3.1.4

Multiply $7$ by $- 1$ .

$y + 1 = 7 x - 7$

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

Subtract $1$ from both sides of the equation.

$y = 7 x - 7 - 1$

Step 3.3.2.2

Subtract $1$ from $- 7$ .

$y = 7 x - 8$

Step 4