Calculus Examples

$y = \frac{5 x}{x - 3}$ , $(4, 20)$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = 20$ to find the slope of the tangent line.

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Step 1.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5 x}{x - 3}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{x}{x - 3}]$ .

$5 \frac{d}{d x} [\frac{x}{x - 3}]$

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 $g (x) = x - 3$ .

$5 \frac{(x - 3) \frac{d}{d x} [x] - x \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

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 $n = 1$ .

$5 \frac{(x - 3) \cdot 1 - x \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 1.3.2

Multiply $x - 3$ by $1$ .

$5 \frac{x - 3 - x \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 1.3.3

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

$5 \frac{x - 3 - x (\frac{d}{d x} [x] + \frac{d}{d x} [- 3])}{{(x - 3)}^{2}}$

Step 1.3.4

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

$5 \frac{x - 3 - x (1 + \frac{d}{d x} [- 3])}{{(x - 3)}^{2}}$

Step 1.3.5

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

$5 \frac{x - 3 - x (1 + 0)}{{(x - 3)}^{2}}$

Step 1.3.6

Simplify terms.

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Step 1.3.6.1

Add $1$ and $0$ .

$5 \frac{x - 3 - x \cdot 1}{{(x - 3)}^{2}}$

Step 1.3.6.2

Multiply $- 1$ by $1$ .

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

Step 1.3.6.3

Subtract $x$ from $x$ .

$5 \frac{0 - 3}{{(x - 3)}^{2}}$

Step 1.3.6.4

Simplify the expression.

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Step 1.3.6.4.1

Subtract $3$ from $0$ .

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

Step 1.3.6.4.2

Move the negative in front of the fraction.

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

Step 1.3.6.4.3

Multiply $- 1$ by $5$ .

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

Step 1.3.6.5

Combine $- 5$ and $\frac{3}{{(x - 3)}^{2}}$ .

$\frac{- 5 \cdot 3}{{(x - 3)}^{2}}$

Step 1.3.6.6

Simplify the expression.

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Step 1.3.6.6.1

Multiply $- 5$ by $3$ .

$\frac{- 15}{{(x - 3)}^{2}}$

Step 1.3.6.6.2

Move the negative in front of the fraction.

$- \frac{15}{{(x - 3)}^{2}}$

Step 1.4

Evaluate the derivative at $x = 4$ .

$- \frac{15}{{((4) - 3)}^{2}}$

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 $4$ .

$- \frac{15}{1^{2}}$

Step 1.5.1.2

One to any power is one.

$- \frac{15}{1}$

Step 1.5.2

Simplify the expression.

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Step 1.5.2.1

Divide $15$ by $1$ .

$- 1 \cdot 15$

Step 1.5.2.2

Multiply $- 1$ by $15$ .

$- 15$

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 $- 15$ and a given point $(4, 20)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (20) = - 15 \cdot (x - (4))$

Step 2.2

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

$y - 20 = - 15 \cdot (x - 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 15 \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

$y - 20 = 0 + 0 - 15 \cdot (x - 4)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 20 = - 15 \cdot (x - 4)$

Step 2.3.1.3

Apply the distributive property.

$y - 20 = - 15 x - 15 \cdot - 4$

Step 2.3.1.4

Multiply $- 15$ by $- 4$ .

$y - 20 = - 15 x + 60$

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 $20$ to both sides of the equation.

$y = - 15 x + 60 + 20$

Step 2.3.2.2

Add $60$ and $20$ .

$y = - 15 x + 80$

Step 3