Calculus Examples

$g (x) = \frac{3 x}{x - 2}$ at $(4, 6)$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = 6$ to find the slope of the tangent line.

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Step 1.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3 x}{x - 2}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{x}{x - 2}]$ .

$3 \frac{d}{d x} [\frac{x}{x - 2}]$

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 $g (x) = x - 2$ .

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

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 $n = 1$ .

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

Step 1.3.2

Multiply $x - 2$ by $1$ .

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

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

$3 \frac{x - 2 - x (1 + 0)}{{(x - 2)}^{2}}$

Step 1.3.6

Simplify terms.

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Step 1.3.6.1

Add $1$ and $0$ .

$3 \frac{x - 2 - x \cdot 1}{{(x - 2)}^{2}}$

Step 1.3.6.2

Multiply $- 1$ by $1$ .

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

Step 1.3.6.3

Subtract $x$ from $x$ .

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

Step 1.3.6.4

Simplify the expression.

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Step 1.3.6.4.1

Subtract $2$ from $0$ .

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

Step 1.3.6.4.2

Move the negative in front of the fraction.

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

Step 1.3.6.4.3

Multiply $- 1$ by $3$ .

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

Step 1.3.6.5

Combine $- 3$ and $\frac{2}{{(x - 2)}^{2}}$ .

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

Step 1.3.6.6

Simplify the expression.

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Step 1.3.6.6.1

Multiply $- 3$ by $2$ .

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

Step 1.3.6.6.2

Move the negative in front of the fraction.

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

Step 1.4

Evaluate the derivative at $x = 4$ .

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

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 $2$ from $4$ .

$- \frac{6}{2^{2}}$

Step 1.5.1.2

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

$- \frac{6}{4}$

Step 1.5.2

Cancel the common factor of $6$ and $4$ .

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Step 1.5.2.1

Factor $2$ out of $6$ .

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

Step 1.5.2.2

Cancel the common factors.

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Step 1.5.2.2.1

Factor $2$ out of $4$ .

$- \frac{2 \cdot 3}{2 \cdot 2}$

Step 1.5.2.2.2

Cancel the common factor.

$- \frac{2 \cdot 3}{2 \cdot 2}$

Step 1.5.2.2.3

Rewrite the expression.

$- \frac{3}{2}$

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 $- \frac{3}{2}$ and a given point $(4, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = - \frac{3}{2} \cdot (x - (4))$

Step 2.2

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

$y - 6 = - \frac{3}{2} \cdot (x - 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

$y - 6 = 0 + 0 - \frac{3}{2} \cdot (x - 4)$

Step 2.3.1.2

Simplify terms.

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Step 2.3.1.2.1

Apply the distributive property.

$y - 6 = - \frac{3}{2} x - \frac{3}{2} \cdot - 4$

Step 2.3.1.2.2

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

$y - 6 = - \frac{x \cdot 3}{2} - \frac{3}{2} \cdot - 4$

Step 2.3.1.2.3

Cancel the common factor of $2$ .

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Step 2.3.1.2.3.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$y - 6 = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot - 4$

Step 2.3.1.2.3.2

Factor $2$ out of $- 4$ .

$y - 6 = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot (2 (- 2))$

Step 2.3.1.2.3.3

Cancel the common factor.

$y - 6 = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot (2 \cdot - 2)$

Step 2.3.1.2.3.4

Rewrite the expression.

$y - 6 = - \frac{x \cdot 3}{2} - 3 \cdot - 2$

Step 2.3.1.2.4

Multiply $- 3$ by $- 2$ .

$y - 6 = - \frac{x \cdot 3}{2} + 6$

Step 2.3.1.3

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

$y - 6 = - \frac{3 x}{2} + 6$

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 $6$ to both sides of the equation.

$y = - \frac{3 x}{2} + 6 + 6$

Step 2.3.2.2

Add $6$ and $6$ .

$y = - \frac{3 x}{2} + 12$

Step 2.3.3

Write in $y = m x + b$ form.

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{3}{2} x) + 12$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{3}{2} x + 12$

Step 3