Calculus Examples

$y = \frac{7 x}{x - 3}$ , $(6, 14)$

Step 1

Find the first derivative and evaluate at $x = 6$ and $y = 14$ 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 $\frac{7 x}{x - 3}$ with respect to $x$ is $7 \frac{d}{d x} [\frac{x}{x - 3}]$ .

$7 \frac{d}{d x} [\frac{x}{x - 3}]$

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 $g (x) = x - 3$ .

$7 \frac{(x - 3) \frac{d}{d x} [x] - x \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

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 $n = 1$ .

$7 \frac{(x - 3) \cdot 1 - x \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 1.3.2

Multiply $x - 3$ by $1$ .

$7 \frac{x - 3 - x \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 1.3.3

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$7 \frac{x - 3 - x (\frac{d}{d x} [x] + \frac{d}{d x} [- 3])}{{(x - 3)}^{2}}$

Step 1.3.4

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

$7 \frac{x - 3 - x (1 + \frac{d}{d x} [- 3])}{{(x - 3)}^{2}}$

Step 1.3.5

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$7 \frac{x - 3 - x (1 + 0)}{{(x - 3)}^{2}}$

Step 1.3.6

Simplify terms.

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Step 1.3.6.1

Add $1$ and $0$ .

$7 \frac{x - 3 - x \cdot 1}{{(x - 3)}^{2}}$

Step 1.3.6.2

Multiply $- 1$ by $1$ .

$7 \frac{x - 3 - x}{{(x - 3)}^{2}}$

Step 1.3.6.3

Subtract $x$ from $x$ .

$7 \frac{0 - 3}{{(x - 3)}^{2}}$

Step 1.3.6.4

Simplify the expression.

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Step 1.3.6.4.1

Subtract $3$ from $0$ .

$7 \frac{- 3}{{(x - 3)}^{2}}$

Step 1.3.6.4.2

Move the negative in front of the fraction.

$7 (- \frac{3}{{(x - 3)}^{2}})$

Step 1.3.6.4.3

Multiply $- 1$ by $7$ .

$- 7 \frac{3}{{(x - 3)}^{2}}$

Step 1.3.6.5

Combine $- 7$ and $\frac{3}{{(x - 3)}^{2}}$ .

$\frac{- 7 \cdot 3}{{(x - 3)}^{2}}$

Step 1.3.6.6

Simplify the expression.

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Step 1.3.6.6.1

Multiply $- 7$ by $3$ .

$\frac{- 21}{{(x - 3)}^{2}}$

Step 1.3.6.6.2

Move the negative in front of the fraction.

$- \frac{21}{{(x - 3)}^{2}}$

Step 1.4

Evaluate the derivative at $x = 6$ .

$- \frac{21}{{((6) - 3)}^{2}}$

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 $3$ from $6$ .

$- \frac{21}{3^{2}}$

Step 1.5.1.2

Raise $3$ to the power of $2$ .

$- \frac{21}{9}$

Step 1.5.2

Cancel the common factor of $21$ and $9$ .

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Step 1.5.2.1

Factor $3$ out of $21$ .

$- \frac{3 (7)}{9}$

Step 1.5.2.2

Cancel the common factors.

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Step 1.5.2.2.1

Factor $3$ out of $9$ .

$- \frac{3 \cdot 7}{3 \cdot 3}$

Step 1.5.2.2.2

Cancel the common factor.

$- \frac{3 \cdot 7}{3 \cdot 3}$

Step 1.5.2.2.3

Rewrite the expression.

$- \frac{7}{3}$

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 $- \frac{7}{3}$ and a given point $(6, 14)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (14) = - \frac{7}{3} \cdot (x - (6))$

Step 2.2

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

$y - 14 = - \frac{7}{3} \cdot (x - 6)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{7}{3} \cdot (x - 6)$ .

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Step 2.3.1.1

Rewrite.

$y - 14 = 0 + 0 - \frac{7}{3} \cdot (x - 6)$

Step 2.3.1.2

Simplify terms.

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Step 2.3.1.2.1

Apply the distributive property.

$y - 14 = - \frac{7}{3} x - \frac{7}{3} \cdot - 6$

Step 2.3.1.2.2

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

$y - 14 = - \frac{x \cdot 7}{3} - \frac{7}{3} \cdot - 6$

Step 2.3.1.2.3

Cancel the common factor of $3$ .

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Step 2.3.1.2.3.1

Move the leading negative in $- \frac{7}{3}$ into the numerator.

$y - 14 = - \frac{x \cdot 7}{3} + \frac{- 7}{3} \cdot - 6$

Step 2.3.1.2.3.2

Factor $3$ out of $- 6$ .

$y - 14 = - \frac{x \cdot 7}{3} + \frac{- 7}{3} \cdot (3 (- 2))$

Step 2.3.1.2.3.3

Cancel the common factor.

$y - 14 = - \frac{x \cdot 7}{3} + \frac{- 7}{3} \cdot (3 \cdot - 2)$

Step 2.3.1.2.3.4

Rewrite the expression.

$y - 14 = - \frac{x \cdot 7}{3} - 7 \cdot - 2$

Step 2.3.1.2.4

Multiply $- 7$ by $- 2$ .

$y - 14 = - \frac{x \cdot 7}{3} + 14$

Step 2.3.1.3

Move $7$ to the left of $x$ .

$y - 14 = - \frac{7 x}{3} + 14$

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.

$y = - \frac{7 x}{3} + 14 + 14$

Step 2.3.2.2

Add $14$ and $14$ .

$y = - \frac{7 x}{3} + 28$

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{7}{3} x) + 28$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{7}{3} x + 28$

Step 3