Calculus Examples

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

Step 1

Find the first derivative and evaluate at $x = 3$ and $y = 3$ to find the slope of the tangent line.

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Step 1.1

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

$7 \frac{d}{d x} [\frac{x}{x + 4}]$

Step 1.2

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 $f (x) = x$ and $g (x) = x + 4$ .

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

Step 1.3

Differentiate.

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Step 1.3.1

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

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

Step 1.3.2

Multiply $x + 4$ by $1$ .

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

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

$7 \frac{x + 4 - x (1 + 0)}{{(x + 4)}^{2}}$

Step 1.3.6

Simplify terms.

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Step 1.3.6.1

Add $1$ and $0$ .

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

Step 1.3.6.2

Multiply $- 1$ by $1$ .

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

Step 1.3.6.3

Subtract $x$ from $x$ .

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

Step 1.3.6.4

Add $0$ and $4$ .

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

Step 1.3.6.5

Combine $7$ and $\frac{4}{{(x + 4)}^{2}}$ .

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

Step 1.3.6.6

Multiply $7$ by $4$ .

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

Step 1.4

Evaluate the derivative at $x = 3$ .

$\frac{28}{{((3) + 4)}^{2}}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the denominator.

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Step 1.5.1.1

Add $3$ and $4$ .

$\frac{28}{7^{2}}$

Step 1.5.1.2

Raise $7$ to the power of $2$ .

$\frac{28}{49}$

Step 1.5.2

Cancel the common factor of $28$ and $49$ .

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Step 1.5.2.1

Factor $7$ out of $28$ .

$\frac{7 (4)}{49}$

Step 1.5.2.2

Cancel the common factors.

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Step 1.5.2.2.1

Factor $7$ out of $49$ .

$\frac{7 \cdot 4}{7 \cdot 7}$

Step 1.5.2.2.2

Cancel the common factor.

$\frac{7 \cdot 4}{7 \cdot 7}$

Step 1.5.2.2.3

Rewrite the expression.

$\frac{4}{7}$

Step 2

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

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Step 2.1

Use the slope $\frac{4}{7}$ and a given point $(3, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = \frac{4}{7} \cdot (x - (3))$

Step 2.2

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

$y - 3 = \frac{4}{7} \cdot (x - 3)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{4}{7} \cdot (x - 3)$ .

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Step 2.3.1.1

Rewrite.

$y - 3 = 0 + 0 + \frac{4}{7} \cdot (x - 3)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 3 = \frac{4}{7} \cdot (x - 3)$

Step 2.3.1.3

Apply the distributive property.

$y - 3 = \frac{4}{7} x + \frac{4}{7} \cdot - 3$

Step 2.3.1.4

Combine $\frac{4}{7}$ and $x$ .

$y - 3 = \frac{4 x}{7} + \frac{4}{7} \cdot - 3$

Step 2.3.1.5

Multiply $\frac{4}{7} \cdot - 3$ .

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Step 2.3.1.5.1

Combine $\frac{4}{7}$ and $- 3$ .

$y - 3 = \frac{4 x}{7} + \frac{4 \cdot - 3}{7}$

Step 2.3.1.5.2

Multiply $4$ by $- 3$ .

$y - 3 = \frac{4 x}{7} + \frac{- 12}{7}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y - 3 = \frac{4 x}{7} - \frac{12}{7}$

Step 2.3.2

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

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Step 2.3.2.1

Add $3$ to both sides of the equation.

$y = \frac{4 x}{7} - \frac{12}{7} + 3$

Step 2.3.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = \frac{4 x}{7} - \frac{12}{7} + 3 \cdot \frac{7}{7}$

Step 2.3.2.3

Combine $3$ and $\frac{7}{7}$ .

$y = \frac{4 x}{7} - \frac{12}{7} + \frac{3 \cdot 7}{7}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{4 x}{7} + \frac{- 12 + 3 \cdot 7}{7}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $3$ by $7$ .

$y = \frac{4 x}{7} + \frac{- 12 + 21}{7}$

Step 2.3.2.5.2

Add $- 12$ and $21$ .

$y = \frac{4 x}{7} + \frac{9}{7}$

Step 2.3.3

Reorder terms.

$y = \frac{4}{7} x + \frac{9}{7}$

Step 3