Calculus Examples

$y = \frac{x + 2}{x - 4}$ , $(3, - 5)$

Step 1

Find the first derivative and evaluate at $x = 3$ and $y = - 5$ to find the slope of the tangent line.

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Step 1.1

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

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{(x - 4) (1 + 0) - (x + 2) \frac{d}{d x} [x - 4]}{{(x - 4)}^{2}}$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $1$ and $0$ .

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

Step 1.2.4.2

Multiply $x - 4$ by $1$ .

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

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

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

Step 1.2.8

Simplify the expression.

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Step 1.2.8.1

Add $1$ and $0$ .

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

Step 1.2.8.2

Multiply $- 1$ by $1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Simplify the numerator.

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Step 1.3.2.1

Combine the opposite terms in $x - 4 - x - 1 \cdot 2$ .

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Step 1.3.2.1.1

Subtract $x$ from $x$ .

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

Step 1.3.2.1.2

Subtract $4$ from $0$ .

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

Step 1.3.2.2

Multiply $- 1$ by $2$ .

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

Step 1.3.2.3

Subtract $2$ from $- 4$ .

$\frac{- 6}{{(x - 4)}^{2}}$

Step 1.3.3

Move the negative in front of the fraction.

$- \frac{6}{{(x - 4)}^{2}}$

Step 1.4

Evaluate the derivative at $x = 3$ .

$- \frac{6}{{((3) - 4)}^{2}}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the denominator.

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Step 1.5.1.1

Subtract $4$ from $3$ .

$- \frac{6}{{(- 1)}^{2}}$

Step 1.5.1.2

Raise $- 1$ to the power of $2$ .

$- \frac{6}{1}$

Step 1.5.2

Simplify the expression.

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Step 1.5.2.1

Divide $6$ by $1$ .

$- 1 \cdot 6$

Step 1.5.2.2

Multiply $- 1$ by $6$ .

$- 6$

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 $- 6$ and a given point $(3, - 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 5) = - 6 \cdot (x - (3))$

Step 2.2

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

$y + 5 = - 6 \cdot (x - 3)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 6 \cdot (x - 3)$ .

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Step 2.3.1.1

Rewrite.

$y + 5 = 0 + 0 - 6 \cdot (x - 3)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 5 = - 6 \cdot (x - 3)$

Step 2.3.1.3

Apply the distributive property.

$y + 5 = - 6 x - 6 \cdot - 3$

Step 2.3.1.4

Multiply $- 6$ by $- 3$ .

$y + 5 = - 6 x + 18$

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 $5$ from both sides of the equation.

$y = - 6 x + 18 - 5$

Step 2.3.2.2

Subtract $5$ from $18$ .

$y = - 6 x + 13$

Step 3