Calculus Examples

$f (x) = \sqrt{5 x + 11}$ , $(5, 6)$

Step 1

Find the first derivative and evaluate at $x = 5$ and $y = 6$ 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 $\sqrt{5 x + 11}$ as ${(5 x + 11)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(5 x + 11)}^{\frac{1}{2}}]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 5 x + 11$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $5 x + 11$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [5 x + 11]$

Step 1.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [5 x + 11]$

Step 1.2.3

Replace all occurrences of $u$ with $5 x + 11$ .

$\frac{1}{2} {(5 x + 11)}^{\frac{1}{2} - 1} \frac{d}{d x} [5 x + 11]$

Step 1.3

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

$\frac{1}{2} {(5 x + 11)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [5 x + 11]$

Step 1.4

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

$\frac{1}{2} {(5 x + 11)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [5 x + 11]$

Step 1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(5 x + 11)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [5 x + 11]$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(5 x + 11)}^{\frac{1 - 2}{2}} \frac{d}{d x} [5 x + 11]$

Step 1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(5 x + 11)}^{\frac{- 1}{2}} \frac{d}{d x} [5 x + 11]$

Step 1.7

Combine fractions.

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Step 1.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(5 x + 11)}^{- \frac{1}{2}} \frac{d}{d x} [5 x + 11]$

Step 1.7.2

Combine $\frac{1}{2}$ and ${(5 x + 11)}^{- \frac{1}{2}}$ .

$\frac{{(5 x + 11)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [5 x + 11]$

Step 1.7.3

Move ${(5 x + 11)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(5 x + 11)}^{\frac{1}{2}}} \frac{d}{d x} [5 x + 11]$

Step 1.8

By the Sum Rule, the derivative of $5 x + 11$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [11]$ .

$\frac{1}{2 {(5 x + 11)}^{\frac{1}{2}}} (\frac{d}{d x} [5 x] + \frac{d}{d x} [11])$

Step 1.9

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\frac{1}{2 {(5 x + 11)}^{\frac{1}{2}}} (5 \frac{d}{d x} [x] + \frac{d}{d x} [11])$

Step 1.10

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

$\frac{1}{2 {(5 x + 11)}^{\frac{1}{2}}} (5 \cdot 1 + \frac{d}{d x} [11])$

Step 1.11

Multiply $5$ by $1$ .

$\frac{1}{2 {(5 x + 11)}^{\frac{1}{2}}} (5 + \frac{d}{d x} [11])$

Step 1.12

Since $11$ is constant with respect to $x$ , the derivative of $11$ with respect to $x$ is $0$ .

$\frac{1}{2 {(5 x + 11)}^{\frac{1}{2}}} (5 + 0)$

Step 1.13

Combine fractions.

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Step 1.13.1

Add $5$ and $0$ .

$\frac{1}{2 {(5 x + 11)}^{\frac{1}{2}}} \cdot 5$

Step 1.13.2

Combine $\frac{1}{2 {(5 x + 11)}^{\frac{1}{2}}}$ and $5$ .

$\frac{5}{2 {(5 x + 11)}^{\frac{1}{2}}}$

Step 1.14

Evaluate the derivative at $x = 5$ .

$\frac{5}{2 {(5 (5) + 11)}^{\frac{1}{2}}}$

Step 1.15

Simplify.

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Step 1.15.1

Simplify the denominator.

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Step 1.15.1.1

Multiply $5$ by $5$ .

$\frac{5}{2 {(25 + 11)}^{\frac{1}{2}}}$

Step 1.15.1.2

Add $25$ and $11$ .

$\frac{5}{2 \cdot 36^{\frac{1}{2}}}$

Step 1.15.1.3

Rewrite $36$ as $6^{2}$ .

$\frac{5}{2 \cdot {(6^{2})}^{\frac{1}{2}}}$

Step 1.15.1.4

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

$\frac{5}{2 \cdot 6^{2 (\frac{1}{2})}}$

Step 1.15.1.5

Cancel the common factor of $2$ .

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Step 1.15.1.5.1

Cancel the common factor.

$\frac{5}{2 \cdot 6^{2 (\frac{1}{2})}}$

Step 1.15.1.5.2

Rewrite the expression.

$\frac{5}{2 \cdot 6^{1}}$

Step 1.15.1.6

Evaluate the exponent.

$\frac{5}{2 \cdot 6}$

Step 1.15.2

Multiply $2$ by $6$ .

$\frac{5}{12}$

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 $\frac{5}{12}$ and a given point $(5, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = \frac{5}{12} \cdot (x - (5))$

Step 2.2

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

$y - 6 = \frac{5}{12} \cdot (x - 5)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{5}{12} \cdot (x - 5)$ .

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Step 2.3.1.1

Rewrite.

$y - 6 = 0 + 0 + \frac{5}{12} \cdot (x - 5)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 6 = \frac{5}{12} \cdot (x - 5)$

Step 2.3.1.3

Apply the distributive property.

$y - 6 = \frac{5}{12} x + \frac{5}{12} \cdot - 5$

Step 2.3.1.4

Combine $\frac{5}{12}$ and $x$ .

$y - 6 = \frac{5 x}{12} + \frac{5}{12} \cdot - 5$

Step 2.3.1.5

Multiply $\frac{5}{12} \cdot - 5$ .

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Step 2.3.1.5.1

Combine $\frac{5}{12}$ and $- 5$ .

$y - 6 = \frac{5 x}{12} + \frac{5 \cdot - 5}{12}$

Step 2.3.1.5.2

Multiply $5$ by $- 5$ .

$y - 6 = \frac{5 x}{12} + \frac{- 25}{12}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y - 6 = \frac{5 x}{12} - \frac{25}{12}$

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 $6$ to both sides of the equation.

$y = \frac{5 x}{12} - \frac{25}{12} + 6$

Step 2.3.2.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$y = \frac{5 x}{12} - \frac{25}{12} + 6 \cdot \frac{12}{12}$

Step 2.3.2.3

Combine $6$ and $\frac{12}{12}$ .

$y = \frac{5 x}{12} - \frac{25}{12} + \frac{6 \cdot 12}{12}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{5 x}{12} + \frac{- 25 + 6 \cdot 12}{12}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $6$ by $12$ .

$y = \frac{5 x}{12} + \frac{- 25 + 72}{12}$

Step 2.3.2.5.2

Add $- 25$ and $72$ .

$y = \frac{5 x}{12} + \frac{47}{12}$

Step 2.3.3

Reorder terms.

$y = \frac{5}{12} x + \frac{47}{12}$

Step 3