Calculus Examples

$g (x) = \frac{4 x}{x - 5}$ at $(7, 14)$

Step 1

Find the first derivative and evaluate at $x = 7$ and $y = 14$ to find the slope of the tangent line.

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Step 1.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4 x}{x - 5}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{x}{x - 5}]$ .

$4 \frac{d}{d x} [\frac{x}{x - 5}]$

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 $g (x) = x - 5$ .

$4 \frac{(x - 5) \frac{d}{d x} [x] - x \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

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 $n = 1$ .

$4 \frac{(x - 5) \cdot 1 - x \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 1.3.2

Multiply $x - 5$ by $1$ .

$4 \frac{x - 5 - x \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 1.3.3

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

$4 \frac{x - 5 - x (\frac{d}{d x} [x] + \frac{d}{d x} [- 5])}{{(x - 5)}^{2}}$

Step 1.3.4

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

$4 \frac{x - 5 - x (1 + \frac{d}{d x} [- 5])}{{(x - 5)}^{2}}$

Step 1.3.5

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

$4 \frac{x - 5 - x (1 + 0)}{{(x - 5)}^{2}}$

Step 1.3.6

Simplify terms.

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Step 1.3.6.1

Add $1$ and $0$ .

$4 \frac{x - 5 - x \cdot 1}{{(x - 5)}^{2}}$

Step 1.3.6.2

Multiply $- 1$ by $1$ .

$4 \frac{x - 5 - x}{{(x - 5)}^{2}}$

Step 1.3.6.3

Subtract $x$ from $x$ .

$4 \frac{0 - 5}{{(x - 5)}^{2}}$

Step 1.3.6.4

Simplify the expression.

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Step 1.3.6.4.1

Subtract $5$ from $0$ .

$4 \frac{- 5}{{(x - 5)}^{2}}$

Step 1.3.6.4.2

Move the negative in front of the fraction.

$4 (- \frac{5}{{(x - 5)}^{2}})$

Step 1.3.6.4.3

Multiply $- 1$ by $4$ .

$- 4 \frac{5}{{(x - 5)}^{2}}$

Step 1.3.6.5

Combine $- 4$ and $\frac{5}{{(x - 5)}^{2}}$ .

$\frac{- 4 \cdot 5}{{(x - 5)}^{2}}$

Step 1.3.6.6

Simplify the expression.

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Step 1.3.6.6.1

Multiply $- 4$ by $5$ .

$\frac{- 20}{{(x - 5)}^{2}}$

Step 1.3.6.6.2

Move the negative in front of the fraction.

$- \frac{20}{{(x - 5)}^{2}}$

Step 1.4

Evaluate the derivative at $x = 7$ .

$- \frac{20}{{((7) - 5)}^{2}}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the denominator.

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Step 1.5.1.1

Subtract $5$ from $7$ .

$- \frac{20}{2^{2}}$

Step 1.5.1.2

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

$- \frac{20}{4}$

Step 1.5.2

Simplify the expression.

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Step 1.5.2.1

Divide $20$ by $4$ .

$- 1 \cdot 5$

Step 1.5.2.2

Multiply $- 1$ by $5$ .

$- 5$

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 $- 5$ and a given point $(7, 14)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (14) = - 5 \cdot (x - (7))$

Step 2.2

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

$y - 14 = - 5 \cdot (x - 7)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 5 \cdot (x - 7)$ .

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Step 2.3.1.1

Rewrite.

$y - 14 = 0 + 0 - 5 \cdot (x - 7)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 14 = - 5 \cdot (x - 7)$

Step 2.3.1.3

Apply the distributive property.

$y - 14 = - 5 x - 5 \cdot - 7$

Step 2.3.1.4

Multiply $- 5$ by $- 7$ .

$y - 14 = - 5 x + 35$

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 $14$ to both sides of the equation.

$y = - 5 x + 35 + 14$

Step 2.3.2.2

Add $35$ and $14$ .

$y = - 5 x + 49$

Step 3