Calculus Examples

$y = \frac{2 x + 1}{x + 2}$ , $(1, 1)$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = 1$ to find the slope of the tangent line.

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Step 1.1

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 $f (x) = 2 x + 1$ and $g (x) = x + 2$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $2 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Multiply $2$ by $1$ .

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

Step 1.2.5

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

$\frac{(x + 2) (2 + 0) - (2 x + 1) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $2$ and $0$ .

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

Step 1.2.6.2

Move $2$ to the left of $x + 2$ .

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

Step 1.2.7

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

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

Step 1.2.8

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

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

Step 1.2.9

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

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

Step 1.2.10

Simplify the expression.

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Step 1.2.10.1

Add $1$ and $0$ .

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

Step 1.2.10.2

Multiply $- 1$ by $1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Simplify the numerator.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Multiply $2$ by $2$ .

$\frac{2 x + 4 - (2 x) - 1 \cdot 1}{{(x + 2)}^{2}}$

Step 1.3.3.1.2

Multiply $2$ by $- 1$ .

$\frac{2 x + 4 - 2 x - 1 \cdot 1}{{(x + 2)}^{2}}$

Step 1.3.3.1.3

Multiply $- 1$ by $1$ .

$\frac{2 x + 4 - 2 x - 1}{{(x + 2)}^{2}}$

Step 1.3.3.2

Combine the opposite terms in $2 x + 4 - 2 x - 1$ .

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Step 1.3.3.2.1

Subtract $2 x$ from $2 x$ .

$\frac{0 + 4 - 1}{{(x + 2)}^{2}}$

Step 1.3.3.2.2

Add $0$ and $4$ .

$\frac{4 - 1}{{(x + 2)}^{2}}$

Step 1.3.3.3

Subtract $1$ from $4$ .

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

Step 1.4

Evaluate the derivative at $x = 1$ .

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

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 $1$ and $2$ .

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

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

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

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

Step 2.3.1.5

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

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

Step 2.3.1.6

Move the negative in front of the fraction.

$y - 1 = \frac{x}{3} - \frac{1}{3}$

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 $1$ to both sides of the equation.

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

Step 2.3.2.2

Write $1$ as a fraction with a common denominator.

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

Step 2.3.2.3

Combine the numerators over the common denominator.

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

Step 2.3.2.4

Add $- 1$ and $3$ .

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

Step 2.3.3

Reorder terms.

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

Step 3