Calculus Examples

$y = \frac{4 x}{x + 2}$ , $(2, 2)$

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 $4$ is constant with respect to $x$ , the derivative of $\frac{4 x}{x + 2}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{x}{x + 2}]$ .

$4 \frac{d}{d x} [\frac{x}{x + 2}]$

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 $g (x) = x + 2$ .

$4 \frac{(x + 2) \frac{d}{d x} [x] - x \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

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 $n = 1$ .

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

Step 1.3.2

Multiply $x + 2$ by $1$ .

$4 \frac{x + 2 - x \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 1.3.3

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

$4 \frac{x + 2 - x (\frac{d}{d x} [x] + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 1.3.4

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

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

Step 1.3.5

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

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

Step 1.3.6

Simplify terms.

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Step 1.3.6.1

Add $1$ and $0$ .

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

Step 1.3.6.2

Multiply $- 1$ by $1$ .

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

Step 1.3.6.3

Subtract $x$ from $x$ .

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

Step 1.3.6.4

Add $0$ and $2$ .

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

Step 1.3.6.5

Combine $4$ and $\frac{2}{{(x + 2)}^{2}}$ .

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

Step 1.3.6.6

Multiply $4$ by $2$ .

$\frac{8}{{(x + 2)}^{2}}$

Step 1.4

Evaluate the derivative at $x = 2$ .

$\frac{8}{{((2) + 2)}^{2}}$

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 $2$ and $2$ .

$\frac{8}{4^{2}}$

Step 1.5.1.2

Raise $4$ to the power of $2$ .

$\frac{8}{16}$

Step 1.5.2

Cancel the common factor of $8$ and $16$ .

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Step 1.5.2.1

Factor $8$ out of $8$ .

$\frac{8 (1)}{16}$

Step 1.5.2.2

Cancel the common factors.

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Step 1.5.2.2.1

Factor $8$ out of $16$ .

$\frac{8 \cdot 1}{8 \cdot 2}$

Step 1.5.2.2.2

Cancel the common factor.

$\frac{8 \cdot 1}{8 \cdot 2}$

Step 1.5.2.2.3

Rewrite the expression.

$\frac{1}{2}$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

Combine $\frac{1}{2}$ and $x$ .

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

Step 2.3.1.5

Cancel the common factor of $2$ .

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Step 2.3.1.5.1

Factor $2$ out of $- 2$ .

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

Step 2.3.1.5.2

Cancel the common factor.

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

Step 2.3.1.5.3

Rewrite the expression.

$y - 2 = \frac{x}{2} - 1$

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.

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

Step 2.3.2.2

Add $- 1$ and $2$ .

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

Step 2.3.3

Reorder terms.

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

Step 3