Calculus Examples

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

Step 1

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

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Step 1.1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6 x}{x + 2}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{x}{x + 2}]$ .

$6 \frac{d}{d x} [\frac{x}{x + 2}]$

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 $g (x) = x + 2$ .

$6 \frac{(x + 2) \frac{d}{d x} [x] - x \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

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 $n = 1$ .

$6 \frac{(x + 2) \cdot 1 - x \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 1.3.2

Multiply $x + 2$ by $1$ .

$6 \frac{x + 2 - x \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 1.3.3

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

$6 \frac{x + 2 - x (\frac{d}{d x} [x] + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 1.3.4

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

$6 \frac{x + 2 - x (1 + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 1.3.5

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

$6 \frac{x + 2 - x (1 + 0)}{{(x + 2)}^{2}}$

Step 1.3.6

Simplify terms.

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Step 1.3.6.1

Add $1$ and $0$ .

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

Step 1.3.6.2

Multiply $- 1$ by $1$ .

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

Step 1.3.6.3

Subtract $x$ from $x$ .

$6 \frac{0 + 2}{{(x + 2)}^{2}}$

Step 1.3.6.4

Add $0$ and $2$ .

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

Step 1.3.6.5

Combine $6$ and $\frac{2}{{(x + 2)}^{2}}$ .

$\frac{6 \cdot 2}{{(x + 2)}^{2}}$

Step 1.3.6.6

Multiply $6$ by $2$ .

$\frac{12}{{(x + 2)}^{2}}$

Step 1.4

Evaluate the derivative at $x = 4$ .

$\frac{12}{{((4) + 2)}^{2}}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the denominator.

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Step 1.5.1.1

Add $4$ and $2$ .

$\frac{12}{6^{2}}$

Step 1.5.1.2

Raise $6$ to the power of $2$ .

$\frac{12}{36}$

Step 1.5.2

Cancel the common factor of $12$ and $36$ .

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Step 1.5.2.1

Factor $12$ out of $12$ .

$\frac{12 (1)}{36}$

Step 1.5.2.2

Cancel the common factors.

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Step 1.5.2.2.1

Factor $12$ out of $36$ .

$\frac{12 \cdot 1}{12 \cdot 3}$

Step 1.5.2.2.2

Cancel the common factor.

$\frac{12 \cdot 1}{12 \cdot 3}$

Step 1.5.2.2.3

Rewrite the expression.

$\frac{1}{3}$

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

$y - (4) = \frac{1}{3} \cdot (x - (4))$

Step 2.2

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

$y - 4 = \frac{1}{3} \cdot (x - 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{1}{3} \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

$y - 4 = 0 + 0 + \frac{1}{3} \cdot (x - 4)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 4 = \frac{1}{3} \cdot (x - 4)$

Step 2.3.1.3

Apply the distributive property.

$y - 4 = \frac{1}{3} x + \frac{1}{3} \cdot - 4$

Step 2.3.1.4

Combine $\frac{1}{3}$ and $x$ .

$y - 4 = \frac{x}{3} + \frac{1}{3} \cdot - 4$

Step 2.3.1.5

Combine $\frac{1}{3}$ and $- 4$ .

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

Step 2.3.1.6

Move the negative in front of the fraction.

$y - 4 = \frac{x}{3} - \frac{4}{3}$

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 $4$ to both sides of the equation.

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

Step 2.3.2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.3.2.3

Combine $4$ and $\frac{3}{3}$ .

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

Step 2.3.2.4

Combine the numerators over the common denominator.

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

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $4$ by $3$ .

$y = \frac{x}{3} + \frac{- 4 + 12}{3}$

Step 2.3.2.5.2

Add $- 4$ and $12$ .

$y = \frac{x}{3} + \frac{8}{3}$

Step 2.3.3

Reorder terms.

$y = \frac{1}{3} x + \frac{8}{3}$

Step 3