Calculus Examples

$g (x) = \frac{6 x}{x + 5}$ at $(- 3, - 9)$

Step 1

Find the first derivative and evaluate at $x = - 3$ and $y = - 9$ to find the slope of the tangent line.

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Step 1.1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6 x}{x + 5}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{x}{x + 5}]$ .

$6 \frac{d}{d x} [\frac{x}{x + 5}]$

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 $g (x) = x + 5$ .

$6 \frac{(x + 5) \frac{d}{d x} [x] - x \frac{d}{d x} [x + 5]}{{(x + 5)}^{2}}$

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 $n = 1$ .

$6 \frac{(x + 5) \cdot 1 - x \frac{d}{d x} [x + 5]}{{(x + 5)}^{2}}$

Step 1.3.2

Multiply $x + 5$ by $1$ .

$6 \frac{x + 5 - x \frac{d}{d x} [x + 5]}{{(x + 5)}^{2}}$

Step 1.3.3

By the Sum Rule, the derivative of $x + 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

$6 \frac{x + 5 - x (\frac{d}{d x} [x] + \frac{d}{d x} [5])}{{(x + 5)}^{2}}$

Step 1.3.4

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

$6 \frac{x + 5 - x (1 + \frac{d}{d x} [5])}{{(x + 5)}^{2}}$

Step 1.3.5

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$6 \frac{x + 5 - x (1 + 0)}{{(x + 5)}^{2}}$

Step 1.3.6

Simplify terms.

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Step 1.3.6.1

Add $1$ and $0$ .

$6 \frac{x + 5 - x \cdot 1}{{(x + 5)}^{2}}$

Step 1.3.6.2

Multiply $- 1$ by $1$ .

$6 \frac{x + 5 - x}{{(x + 5)}^{2}}$

Step 1.3.6.3

Subtract $x$ from $x$ .

$6 \frac{0 + 5}{{(x + 5)}^{2}}$

Step 1.3.6.4

Add $0$ and $5$ .

$6 \frac{5}{{(x + 5)}^{2}}$

Step 1.3.6.5

Combine $6$ and $\frac{5}{{(x + 5)}^{2}}$ .

$\frac{6 \cdot 5}{{(x + 5)}^{2}}$

Step 1.3.6.6

Multiply $6$ by $5$ .

$\frac{30}{{(x + 5)}^{2}}$

Step 1.4

Evaluate the derivative at $x = - 3$ .

$\frac{30}{{((- 3) + 5)}^{2}}$

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 $5$ .

$\frac{30}{2^{2}}$

Step 1.5.1.2

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

$\frac{30}{4}$

Step 1.5.2

Cancel the common factor of $30$ and $4$ .

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Step 1.5.2.1

Factor $2$ out of $30$ .

$\frac{2 (15)}{4}$

Step 1.5.2.2

Cancel the common factors.

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Step 1.5.2.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 15}{2 \cdot 2}$

Step 1.5.2.2.2

Cancel the common factor.

$\frac{2 \cdot 15}{2 \cdot 2}$

Step 1.5.2.2.3

Rewrite the expression.

$\frac{15}{2}$

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 $\frac{15}{2}$ and a given point $(- 3, - 9)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 9) = \frac{15}{2} \cdot (x - (- 3))$

Step 2.2

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

$y + 9 = \frac{15}{2} \cdot (x + 3)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{15}{2} \cdot (x + 3)$ .

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Step 2.3.1.1

Rewrite.

$y + 9 = 0 + 0 + \frac{15}{2} \cdot (x + 3)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 9 = \frac{15}{2} \cdot (x + 3)$

Step 2.3.1.3

Apply the distributive property.

$y + 9 = \frac{15}{2} x + \frac{15}{2} \cdot 3$

Step 2.3.1.4

Combine $\frac{15}{2}$ and $x$ .

$y + 9 = \frac{15 x}{2} + \frac{15}{2} \cdot 3$

Step 2.3.1.5

Multiply $\frac{15}{2} \cdot 3$ .

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Step 2.3.1.5.1

Combine $\frac{15}{2}$ and $3$ .

$y + 9 = \frac{15 x}{2} + \frac{15 \cdot 3}{2}$

Step 2.3.1.5.2

Multiply $15$ by $3$ .

$y + 9 = \frac{15 x}{2} + \frac{45}{2}$

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

Subtract $9$ from both sides of the equation.

$y = \frac{15 x}{2} + \frac{45}{2} - 9$

Step 2.3.2.2

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{15 x}{2} + \frac{45}{2} - 9 \cdot \frac{2}{2}$

Step 2.3.2.3

Combine $- 9$ and $\frac{2}{2}$ .

$y = \frac{15 x}{2} + \frac{45}{2} + \frac{- 9 \cdot 2}{2}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{15 x}{2} + \frac{45 - 9 \cdot 2}{2}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $- 9$ by $2$ .

$y = \frac{15 x}{2} + \frac{45 - 18}{2}$

Step 2.3.2.5.2

Subtract $18$ from $45$ .

$y = \frac{15 x}{2} + \frac{27}{2}$

Step 2.3.3

Reorder terms.

$y = \frac{15}{2} x + \frac{27}{2}$

Step 3