Calculus Examples

$g (x) = \frac{3 x}{x + 1}$ at $(2, 2)$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = 2$ 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 $\frac{3 x}{x + 1}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{x}{x + 1}]$ .

$3 \frac{d}{d x} [\frac{x}{x + 1}]$

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 $g (x) = x + 1$ .

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

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 $n = 1$ .

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

Step 1.3.2

Multiply $x + 1$ by $1$ .

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

Step 1.3.3

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

Step 1.3.4

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

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

Step 1.3.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.3.6

Simplify terms.

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Step 1.3.6.1

Add $1$ and $0$ .

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

Step 1.3.6.2

Multiply $- 1$ by $1$ .

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

Step 1.3.6.3

Subtract $x$ from $x$ .

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

Step 1.3.6.4

Add $0$ and $1$ .

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

Step 1.3.6.5

Combine $3$ and $\frac{1}{{(x + 1)}^{2}}$ .

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

Step 1.4

Evaluate the derivative at $x = 2$ .

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

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 $2$ and $1$ .

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

Step 1.5.1.2

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

$\frac{3}{9}$

Step 1.5.2

Cancel the common factor of $3$ and $9$ .

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Step 1.5.2.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{9}$

Step 1.5.2.2

Cancel the common factors.

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Step 1.5.2.2.1

Factor $3$ out of $9$ .

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

Step 1.5.2.2.2

Cancel the common factor.

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

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 $(2, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

$y - 2 = 0 + 0 + \frac{1}{3} \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

$y - 2 = \frac{1}{3} x + \frac{1}{3} \cdot - 2$

Step 2.3.1.4

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

$y - 2 = \frac{x}{3} + \frac{1}{3} \cdot - 2$

Step 2.3.1.5

Combine $\frac{1}{3}$ and $- 2$ .

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

Step 2.3.1.6

Move the negative in front of the fraction.

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

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 $2$ to both sides of the equation.

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

Step 2.3.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.3.2.3

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

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

Step 2.3.2.4

Combine the numerators over the common denominator.

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

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $2$ by $3$ .

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

Step 2.3.2.5.2

Add $- 2$ and $6$ .

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

Step 2.3.3

Reorder terms.

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

Step 3