Calculus Examples

$y = \frac{\sqrt{x}}{x + 6}$ , $(4, 0.2)$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = 0.2$ to find the slope of the tangent line.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

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 $f (x) = x^{\frac{1}{2}}$ and $g (x) = x + 6$ .

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

Step 1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

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

Step 1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.5

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

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.7.2

Subtract $2$ from $1$ .

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

Step 1.8

Combine fractions.

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Step 1.8.1

Move the negative in front of the fraction.

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

Step 1.8.2

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

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

Step 1.8.3

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.9

By the Sum Rule, the derivative of $x + 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [6]$ .

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

Step 1.10

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

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

Step 1.11

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

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

Step 1.12

Simplify the expression.

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Step 1.12.1

Add $1$ and $0$ .

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

Step 1.12.2

Multiply $- 1$ by $1$ .

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

Step 1.13

Simplify.

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Step 1.13.1

Apply the distributive property.

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

Step 1.13.2

Simplify the numerator.

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Step 1.13.2.1

Simplify each term.

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Step 1.13.2.1.1

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

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

Step 1.13.2.1.2

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 1.13.2.1.3

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 1.13.2.1.3.1

Multiply $x$ by $x^{- \frac{1}{2}}$ .

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Step 1.13.2.1.3.1.1

Raise $x$ to the power of $1$ .

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

Step 1.13.2.1.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.13.2.1.3.2

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

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

Step 1.13.2.1.3.3

Combine the numerators over the common denominator.

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

Step 1.13.2.1.3.4

Subtract $1$ from $2$ .

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

Step 1.13.2.1.4

Cancel the common factor of $2$ .

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Step 1.13.2.1.4.1

Factor $2$ out of $6$ .

$\frac{\frac{x^{\frac{1}{2}}}{2} + 2 \cdot 3 \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}}}{{(x + 6)}^{2}}$

Step 1.13.2.1.4.2

Cancel the common factor.

$\frac{\frac{x^{\frac{1}{2}}}{2} + 2 \cdot 3 \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}}}{{(x + 6)}^{2}}$

Step 1.13.2.1.4.3

Rewrite the expression.

$\frac{\frac{x^{\frac{1}{2}}}{2} + 3 \frac{1}{x^{\frac{1}{2}}} - x^{\frac{1}{2}}}{{(x + 6)}^{2}}$

Step 1.13.2.1.5

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

$\frac{\frac{x^{\frac{1}{2}}}{2} + \frac{3}{x^{\frac{1}{2}}} - x^{\frac{1}{2}}}{{(x + 6)}^{2}}$

Step 1.13.2.2

To write $- x^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{x^{\frac{1}{2}}}{2} - x^{\frac{1}{2}} \cdot \frac{2}{2} + \frac{3}{x^{\frac{1}{2}}}}{{(x + 6)}^{2}}$

Step 1.13.2.3

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

$\frac{\frac{x^{\frac{1}{2}}}{2} + \frac{- x^{\frac{1}{2}} \cdot 2}{2} + \frac{3}{x^{\frac{1}{2}}}}{{(x + 6)}^{2}}$

Step 1.13.2.4

Combine the numerators over the common denominator.

$\frac{\frac{x^{\frac{1}{2}} - x^{\frac{1}{2}} \cdot 2}{2} + \frac{3}{x^{\frac{1}{2}}}}{{(x + 6)}^{2}}$

Step 1.13.2.5

Simplify each term.

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Step 1.13.2.5.1

Simplify the numerator.

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Step 1.13.2.5.1.1

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} - x^{\frac{1}{2}} \cdot 2$ .

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Step 1.13.2.5.1.1.1

Move $x^{\frac{1}{2}}$ .

$\frac{\frac{x^{\frac{1}{2}} - 1 \cdot 2 x^{\frac{1}{2}}}{2} + \frac{3}{x^{\frac{1}{2}}}}{{(x + 6)}^{2}}$

Step 1.13.2.5.1.1.2

Multiply by $1$ .

$\frac{\frac{x^{\frac{1}{2}} \cdot 1 - 1 \cdot 2 x^{\frac{1}{2}}}{2} + \frac{3}{x^{\frac{1}{2}}}}{{(x + 6)}^{2}}$

Step 1.13.2.5.1.1.3

Factor $x^{\frac{1}{2}}$ out of $- 1 \cdot 2 x^{\frac{1}{2}}$ .

$\frac{\frac{x^{\frac{1}{2}} \cdot 1 + x^{\frac{1}{2}} (- 1 \cdot 2)}{2} + \frac{3}{x^{\frac{1}{2}}}}{{(x + 6)}^{2}}$

Step 1.13.2.5.1.1.4

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} \cdot 1 + x^{\frac{1}{2}} (- 1 \cdot 2)$ .

$\frac{\frac{x^{\frac{1}{2}} (1 - 1 \cdot 2)}{2} + \frac{3}{x^{\frac{1}{2}}}}{{(x + 6)}^{2}}$

Step 1.13.2.5.1.2

Multiply $- 1$ by $2$ .

$\frac{\frac{x^{\frac{1}{2}} (1 - 2)}{2} + \frac{3}{x^{\frac{1}{2}}}}{{(x + 6)}^{2}}$

Step 1.13.2.5.1.3

Subtract $2$ from $1$ .

$\frac{\frac{x^{\frac{1}{2}} \cdot - 1}{2} + \frac{3}{x^{\frac{1}{2}}}}{{(x + 6)}^{2}}$

Step 1.13.2.5.2

Move $- 1$ to the left of $x^{\frac{1}{2}}$ .

$\frac{\frac{- 1 \cdot x^{\frac{1}{2}}}{2} + \frac{3}{x^{\frac{1}{2}}}}{{(x + 6)}^{2}}$

Step 1.13.2.5.3

Move the negative in front of the fraction.

$\frac{- \frac{x^{\frac{1}{2}}}{2} + \frac{3}{x^{\frac{1}{2}}}}{{(x + 6)}^{2}}$

Step 1.13.3

Combine terms.

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Step 1.13.3.1

Multiply $\frac{- \frac{x^{\frac{1}{2}}}{2} + \frac{3}{x^{\frac{1}{2}}}}{{(x + 6)}^{2}}$ by $\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$ .

$\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}} \cdot \frac{- \frac{x^{\frac{1}{2}}}{2} + \frac{3}{x^{\frac{1}{2}}}}{{(x + 6)}^{2}}$

Step 1.13.3.2

Combine.

$\frac{x^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2} + \frac{3}{x^{\frac{1}{2}}})}{x^{\frac{1}{2}} {(x + 6)}^{2}}$

Step 1.13.3.3

Apply the distributive property.

$\frac{x^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2}) + x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}}}{x^{\frac{1}{2}} {(x + 6)}^{2}}$

Step 1.13.3.4

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 1.13.3.4.1

Cancel the common factor.

$\frac{x^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2}) + x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}}}{x^{\frac{1}{2}} {(x + 6)}^{2}}$

Step 1.13.3.4.2

Rewrite the expression.

$\frac{x^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2}) + 3}{x^{\frac{1}{2}} {(x + 6)}^{2}}$

Step 1.13.3.5

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

$\frac{- \frac{x^{\frac{1}{2}} x^{\frac{1}{2}}}{2} + 3}{x^{\frac{1}{2}} {(x + 6)}^{2}}$

Step 1.13.3.6

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 1.13.3.6.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- \frac{x^{\frac{1}{2} + \frac{1}{2}}}{2} + 3}{x^{\frac{1}{2}} {(x + 6)}^{2}}$

Step 1.13.3.6.2

Combine the numerators over the common denominator.

$\frac{- \frac{x^{\frac{1 + 1}{2}}}{2} + 3}{x^{\frac{1}{2}} {(x + 6)}^{2}}$

Step 1.13.3.6.3

Add $1$ and $1$ .

$\frac{- \frac{x^{\frac{2}{2}}}{2} + 3}{x^{\frac{1}{2}} {(x + 6)}^{2}}$

Step 1.13.3.6.4

Divide $2$ by $2$ .

$\frac{- \frac{x^{1}}{2} + 3}{x^{\frac{1}{2}} {(x + 6)}^{2}}$

Step 1.13.3.7

Simplify $x^{1}$ .

$\frac{- \frac{x}{2} + 3}{x^{\frac{1}{2}} {(x + 6)}^{2}}$

Step 1.13.4

Simplify the numerator.

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Step 1.13.4.1

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

$\frac{- \frac{x}{2} + 3 \cdot \frac{2}{2}}{x^{\frac{1}{2}} {(x + 6)}^{2}}$

Step 1.13.4.2

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

$\frac{- \frac{x}{2} + \frac{3 \cdot 2}{2}}{x^{\frac{1}{2}} {(x + 6)}^{2}}$

Step 1.13.4.3

Combine the numerators over the common denominator.

$\frac{\frac{- x + 3 \cdot 2}{2}}{x^{\frac{1}{2}} {(x + 6)}^{2}}$

Step 1.13.4.4

Multiply $3$ by $2$ .

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

Step 1.13.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.13.6

Multiply $\frac{- x + 6}{2}$ by $\frac{1}{x^{\frac{1}{2}} {(x + 6)}^{2}}$ .

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

Step 1.13.7

Factor $- 1$ out of $- x$ .

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

Step 1.13.8

Rewrite $6$ as $- 1 (- 6)$ .

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

Step 1.13.9

Factor $- 1$ out of $- (x) - 1 (- 6)$ .

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

Step 1.13.10

Rewrite $- (x - 6)$ as $- 1 (x - 6)$ .

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

Step 1.13.11

Move the negative in front of the fraction.

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

Step 1.14

Evaluate the derivative at $x = 4$ .

$- \frac{(4) - 6}{2 {(4)}^{\frac{1}{2}} {((4) + 6)}^{2}}$

Step 1.15

Simplify.

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Step 1.15.1

Factor $2$ out of $4$ .

$- \frac{2 \cdot 2 - 6}{2 {(4)}^{\frac{1}{2}} {((4) + 6)}^{2}}$

Step 1.15.2

Factor $2$ out of $- 6$ .

$- \frac{2 \cdot 2 + 2 \cdot - 3}{2 {(4)}^{\frac{1}{2}} {((4) + 6)}^{2}}$

Step 1.15.3

Factor $2$ out of $2 \cdot 2 + 2 \cdot - 3$ .

$- \frac{2 \cdot (2 - 3)}{2 {(4)}^{\frac{1}{2}} {((4) + 6)}^{2}}$

Step 1.15.4

Cancel the common factors.

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Step 1.15.4.1

Factor $2$ out of $2 {(4)}^{\frac{1}{2}} {((4) + 6)}^{2}$ .

$- \frac{2 \cdot (2 - 3)}{2 ({(4)}^{\frac{1}{2}} {((4) + 6)}^{2})}$

Step 1.15.4.2

Cancel the common factor.

$- \frac{2 \cdot (2 - 3)}{2 ({(4)}^{\frac{1}{2}} {((4) + 6)}^{2})}$

Step 1.15.4.3

Rewrite the expression.

$- \frac{2 - 3}{{(4)}^{\frac{1}{2}} {((4) + 6)}^{2}}$

Step 1.15.5

Subtract $3$ from $2$ .

$- \frac{- 1}{4^{\frac{1}{2}} {(4 + 6)}^{2}}$

Step 1.15.6

Simplify the denominator.

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Step 1.15.6.1

Add $4$ and $6$ .

$- \frac{- 1}{4^{\frac{1}{2}} \cdot 10^{2}}$

Step 1.15.6.2

Rewrite $4$ as $2^{2}$ .

$- \frac{- 1}{{(2^{2})}^{\frac{1}{2}} \cdot 10^{2}}$

Step 1.15.6.3

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{- 1}{2^{2 (\frac{1}{2})} \cdot 10^{2}}$

Step 1.15.6.4

Cancel the common factor of $2$ .

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Step 1.15.6.4.1

Cancel the common factor.

$- \frac{- 1}{2^{2 (\frac{1}{2})} \cdot 10^{2}}$

Step 1.15.6.4.2

Rewrite the expression.

$- \frac{- 1}{2^{1} \cdot 10^{2}}$

Step 1.15.6.5

Evaluate the exponent.

$- \frac{- 1}{2 \cdot 10^{2}}$

Step 1.15.6.6

Raise $10$ to the power of $2$ .

$- \frac{- 1}{2 \cdot 100}$

Step 1.15.7

Simplify the expression.

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Step 1.15.7.1

Multiply $2$ by $100$ .

$- \frac{- 1}{200}$

Step 1.15.7.2

Move the negative in front of the fraction.

$- - \frac{1}{200}$

Step 1.15.8

Multiply $- - \frac{1}{200}$ .

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Step 1.15.8.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{200})$

Step 1.15.8.2

Multiply $\frac{1}{200}$ by $1$ .

$\frac{1}{200}$

Step 2

The normal line is perpendicular to the tangent line. Take the negative reciprocal of the slope of the tangent line to find the slope of the normal line.

$- 200$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $- 200$ and a given point $(4, 0.2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0.2) = - 200 \cdot (x - (4))$

Step 3.2

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

$y - 0.2 = - 200 \cdot (x - 4)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- 200 \cdot (x - 4)$ .

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Step 3.3.1.1

Rewrite.

$y - 0.2 = 0 + 0 - 200 \cdot (x - 4)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 0.2 = - 200 \cdot (x - 4)$

Step 3.3.1.3

Apply the distributive property.

$y - 0.2 = - 200 x - 200 \cdot - 4$

Step 3.3.1.4

Multiply $- 200$ by $- 4$ .

$y - 0.2 = - 200 x + 800$

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.2.1

Add $0.2$ to both sides of the equation.

$y = - 200 x + 800 + 0.2$

Step 3.3.2.2

Add $800$ and $0.2$ .

$y = - 200 x + 800.2$

Step 4