Calculus Examples

y = \frac{\sqrt{x}}{x + 3}

$y = \frac{\sqrt{x}}{x + 3}$ ,

$(1, 0.25)$

Step 1

Find the first derivative and evaluate at

$x = 1$ and

$y = 0.25$ 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

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

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

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

$g (x) = x + 3$ .

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

Step 1.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = \frac{1}{2}$ .

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

Step 1.4

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 1.5

Combine

$- 1$ and

$\frac{2}{2}$ .

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply

$- 1$ by

$2$ .

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

Step 1.7.2

Subtract

$2$ from

$1$ .

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

Step 1.8

Combine fractions.

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Step 1.8.1

Move the negative in front of the fraction.

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

Step 1.8.2

Combine

$\frac{1}{2}$ and

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

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

Step 1.8.3

Move

$x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.9

By the Sum Rule, the derivative of

$x + 3$ with respect to

$x$ is

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

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

Step 1.10

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.11

Since

$3$ is constant with respect to

$x$ , the derivative of

$3$ with respect to

$x$ is

$0$ .

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

Step 1.12

Simplify the expression.

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Step 1.12.1

Add

$1$ and

$0$ .

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

Step 1.12.2

Multiply

$- 1$ by

$1$ .

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

Step 1.13

Simplify.

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Step 1.13.1

Apply the distributive property.

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

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

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

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

Step 1.13.2.1.2

Move

$x^{\frac{1}{2}}$ to the numerator using the negative exponent rule

$\frac{1}{b^{n}} = b^{- n}$ .

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

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

$1$ .

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

Step 1.13.2.1.3.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.13.2.1.3.2

Write

$1$ as a fraction with a common denominator.

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

Step 1.13.2.1.3.3

Combine the numerators over the common denominator.

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

Step 1.13.2.1.3.4

Subtract

$1$ from

$2$ .

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

Step 1.13.2.1.4

Combine

$3$ and

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

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

Step 1.13.2.2

To write

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

$\frac{2}{2}$ .

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

Step 1.13.2.3

Combine

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

$\frac{2}{2}$ .

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

Step 1.13.2.4

Combine the numerators over the common denominator.

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

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

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

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

Step 1.13.2.5.1.1.2

Multiply by

$1$ .

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

Step 1.13.2.5.1.1.3

Factor

$x^{\frac{1}{2}}$ out of

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

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

Step 1.13.2.5.1.1.4

Factor

$x^{\frac{1}{2}}$ out of

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

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

Step 1.13.2.5.1.2

Multiply

$- 1$ by

$2$ .

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

Step 1.13.2.5.1.3

Subtract

$2$ from

$1$ .

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

Step 1.13.2.5.2

Move

$- 1$ to the left of

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

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

Step 1.13.2.5.3

Move the negative in front of the fraction.

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

Step 1.13.3

Combine terms.

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Step 1.13.3.1

Multiply

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

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

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

Step 1.13.3.2

Combine.

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

Step 1.13.3.3

Apply the distributive property.

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

Step 1.13.3.4

Cancel the common factor of

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

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Step 1.13.3.4.1

Cancel the common factor.

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

Step 1.13.3.4.2

Rewrite the expression.

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

Step 1.13.3.5

Multiply

$- 1$ by

$2$ .

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

Step 1.13.3.6

Combine

$- 2$ and

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

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

Step 1.13.3.7

Combine

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

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

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

Step 1.13.3.8

Multiply

$x^{\frac{1}{2}}$ by

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

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Step 1.13.3.8.1

Move

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

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

Step 1.13.3.8.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.13.3.8.3

Combine the numerators over the common denominator.

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

Step 1.13.3.8.4

Add

$1$ and

$1$ .

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

Step 1.13.3.8.5

Divide

$2$ by

$2$ .

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

Step 1.13.3.9

Simplify

$x^{1} \cdot - 2$ .

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

Step 1.13.3.10

Move

$- 2$ to the left of

$x$ .

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

Step 1.13.3.11

Cancel the common factor of

$- 2$ and

$2$ .

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Step 1.13.3.11.1

Factor

$2$ out of

$- 2 x$ .

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

Step 1.13.3.11.2

Cancel the common factors.

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Step 1.13.3.11.2.1

Factor

$2$ out of

$2$ .

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

Step 1.13.3.11.2.2

Cancel the common factor.

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

Step 1.13.3.11.2.3

Rewrite the expression.

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

Step 1.13.3.11.2.4

Divide

$- x$ by

$1$ .

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

Step 1.13.4

Factor

$- 1$ out of

$- x$ .

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

Step 1.13.5

Rewrite

$3$ as

$- 1 (- 3)$ .

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

Step 1.13.6

Factor

$- 1$ out of

$- (x) - 1 (- 3)$ .

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

Step 1.13.7

Rewrite

$- (x - 3)$ as

$- 1 (x - 3)$ .

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

Step 1.13.8

Move the negative in front of the fraction.

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

Step 1.14

Evaluate the derivative at

$x = 1$ .

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

Step 1.15

Simplify.

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Step 1.15.1

Subtract

$3$ from

$1$ .

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

Step 1.15.2

Simplify the denominator.

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Step 1.15.2.1

Add

$1$ and

$3$ .

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

Step 1.15.2.2

Combine exponents.

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Step 1.15.2.2.1

Rewrite

$4$ as

$2^{2}$ .

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

Step 1.15.2.2.2

Multiply the exponents in

${(2^{2})}^{2}$ .

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Step 1.15.2.2.2.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.15.2.2.2.2

Multiply

$2$ by

$2$ .

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

Step 1.15.2.2.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.15.2.2.4

Add

$4$ and

$1$ .

$- \frac{- 2}{2^{5} \cdot 1^{\frac{1}{2}}}$

Step 1.15.2.3

Raise

$2$ to the power of

$5$ .

$- \frac{- 2}{32 \cdot 1^{\frac{1}{2}}}$

Step 1.15.2.4

One to any power is one.

$- \frac{- 2}{32 \cdot 1}$

Step 1.15.3

Reduce the expression by cancelling the common factors.

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Step 1.15.3.1

Multiply

$32$ by

$1$ .

$- \frac{- 2}{32}$

Step 1.15.3.2

Cancel the common factor of

$- 2$ and

$32$ .

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Step 1.15.3.2.1

Factor

$2$ out of

$- 2$ .

$- \frac{2 (- 1)}{32}$

Step 1.15.3.2.2

Cancel the common factors.

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Step 1.15.3.2.2.1

Factor

$2$ out of

$32$ .

$- \frac{2 \cdot - 1}{2 \cdot 16}$

Step 1.15.3.2.2.2

Cancel the common factor.

$- \frac{2 \cdot - 1}{2 \cdot 16}$

Step 1.15.3.2.2.3

Rewrite the expression.

$- \frac{- 1}{16}$

Step 1.15.3.3

Move the negative in front of the fraction.

$- - \frac{1}{16}$

Step 1.15.4

Multiply

$- - \frac{1}{16}$ .

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Step 1.15.4.1

Multiply

$- 1$ by

$- 1$ .

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

Step 1.15.4.2

Multiply

$\frac{1}{16}$ by

$1$ .

$\frac{1}{16}$

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

$\frac{1}{16}$ and a given point

$(1, 0.25)$ 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

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

$y - (0.25) = \frac{1}{16} \cdot (x - (1))$

Step 2.2

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

$y - 0.25 = \frac{1}{16} \cdot (x - 1)$

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Simplify

$\frac{1}{16} \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 0.25 = 0 + 0 + \frac{1}{16} \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 0.25 = \frac{1}{16} \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 0.25 = \frac{1}{16} x + \frac{1}{16} \cdot - 1$

Step 2.3.1.4

Combine

$\frac{1}{16}$ and

$x$ .

$y - 0.25 = \frac{x}{16} + \frac{1}{16} \cdot - 1$

Step 2.3.1.5

Combine

$\frac{1}{16}$ and

$- 1$ .

$y - 0.25 = \frac{x}{16} + \frac{- 1}{16}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y - 0.25 = \frac{x}{16} - \frac{1}{16}$

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

$0.25$ to both sides of the equation.

$y = \frac{x}{16} - \frac{1}{16} + 0.25$

Step 2.3.2.2

To write

$0.25$ as a fraction with a common denominator, multiply by

$\frac{16}{16}$ .

$y = \frac{x}{16} - \frac{1}{16} + 0.25 \cdot \frac{16}{16}$

Step 2.3.2.3

Combine

$0.25$ and

$\frac{16}{16}$ .

$y = \frac{x}{16} - \frac{1}{16} + \frac{0.25 \cdot 16}{16}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{x}{16} + \frac{- 1 + 0.25 \cdot 16}{16}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply

$0.25$ by

$16$ .

$y = \frac{x}{16} + \frac{- 1 + 4}{16}$

Step 2.3.2.5.2

Add

$- 1$ and

$4$ .

$y = \frac{x}{16} + \frac{3}{16}$

Step 2.3.2.6

Cancel the common factor of

$3$ and

$16$ .

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Step 2.3.2.6.1

Rewrite

$3$ as

$1 (3)$ .

$y = \frac{x}{16} + \frac{1 (3)}{16}$

Step 2.3.2.6.2

Cancel the common factors.

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Step 2.3.2.6.2.1

Rewrite

$16$ as

$1 (16)$ .

$y = \frac{x}{16} + \frac{1 \cdot 3}{1 \cdot 16}$

Step 2.3.2.6.2.2

Cancel the common factor.

$y = \frac{x}{16} + \frac{1 \cdot 3}{1 \cdot 16}$

Step 2.3.2.6.2.3

Rewrite the expression.

$y = \frac{x}{16} + \frac{3}{16}$

Step 2.3.3

Reorder terms.

$y = \frac{1}{16} x + \frac{3}{16}$

Step 3