Calculus Examples

$f (x) = \sqrt{3 x + 16}$ ,

$(3, 5)$

Step 1

Find the first derivative and evaluate at

$x = 3$ and

$y = 5$ 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

$\sqrt{3 x + 16}$ as

${(3 x + 16)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(3 x + 16)}^{\frac{1}{2}}]$

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

$g (x) = 3 x + 16$ .

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Step 1.2.1

To apply the Chain Rule, set

$u$ as

$3 x + 16$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [3 x + 16]$

Step 1.2.2

Differentiate using the Power Rule which states that

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

$n u^{n - 1}$ where

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [3 x + 16]$

Step 1.2.3

Replace all occurrences of

$u$ with

$3 x + 16$ .

$\frac{1}{2} {(3 x + 16)}^{\frac{1}{2} - 1} \frac{d}{d x} [3 x + 16]$

Step 1.3

To write

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

$\frac{2}{2}$ .

$\frac{1}{2} {(3 x + 16)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [3 x + 16]$

Step 1.4

Combine

$- 1$ and

$\frac{2}{2}$ .

$\frac{1}{2} {(3 x + 16)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [3 x + 16]$

Step 1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(3 x + 16)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [3 x + 16]$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply

$- 1$ by

$2$ .

$\frac{1}{2} {(3 x + 16)}^{\frac{1 - 2}{2}} \frac{d}{d x} [3 x + 16]$

Step 1.6.2

Subtract

$2$ from

$1$ .

$\frac{1}{2} {(3 x + 16)}^{\frac{- 1}{2}} \frac{d}{d x} [3 x + 16]$

Step 1.7

Combine fractions.

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Step 1.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(3 x + 16)}^{- \frac{1}{2}} \frac{d}{d x} [3 x + 16]$

Step 1.7.2

Combine

$\frac{1}{2}$ and

${(3 x + 16)}^{- \frac{1}{2}}$ .

$\frac{{(3 x + 16)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [3 x + 16]$

Step 1.7.3

Move

${(3 x + 16)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

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

$\frac{1}{2 {(3 x + 16)}^{\frac{1}{2}}} \frac{d}{d x} [3 x + 16]$

Step 1.8

By the Sum Rule, the derivative of

$3 x + 16$ with respect to

$x$ is

$\frac{d}{d x} [3 x] + \frac{d}{d x} [16]$ .

$\frac{1}{2 {(3 x + 16)}^{\frac{1}{2}}} (\frac{d}{d x} [3 x] + \frac{d}{d x} [16])$

Step 1.9

Since

$3$ is constant with respect to

$x$ , the derivative of

$3 x$ with respect to

$x$ is

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

$\frac{1}{2 {(3 x + 16)}^{\frac{1}{2}}} (3 \frac{d}{d x} [x] + \frac{d}{d x} [16])$

Step 1.10

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

$\frac{1}{2 {(3 x + 16)}^{\frac{1}{2}}} (3 \cdot 1 + \frac{d}{d x} [16])$

Step 1.11

Multiply

$3$ by

$1$ .

$\frac{1}{2 {(3 x + 16)}^{\frac{1}{2}}} (3 + \frac{d}{d x} [16])$

Step 1.12

Since

$16$ is constant with respect to

$x$ , the derivative of

$16$ with respect to

$x$ is

$0$ .

$\frac{1}{2 {(3 x + 16)}^{\frac{1}{2}}} (3 + 0)$

Step 1.13

Combine fractions.

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Step 1.13.1

Add

$3$ and

$0$ .

$\frac{1}{2 {(3 x + 16)}^{\frac{1}{2}}} \cdot 3$

Step 1.13.2

Combine

$\frac{1}{2 {(3 x + 16)}^{\frac{1}{2}}}$ and

$3$ .

$\frac{3}{2 {(3 x + 16)}^{\frac{1}{2}}}$

Step 1.14

Evaluate the derivative at

$x = 3$ .

$\frac{3}{2 {(3 (3) + 16)}^{\frac{1}{2}}}$

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

$3$ by

$3$ .

$\frac{3}{2 {(9 + 16)}^{\frac{1}{2}}}$

Step 1.15.1.2

Add

$9$ and

$16$ .

$\frac{3}{2 \cdot 25^{\frac{1}{2}}}$

Step 1.15.1.3

Rewrite

$25$ as

$5^{2}$ .

$\frac{3}{2 \cdot {(5^{2})}^{\frac{1}{2}}}$

Step 1.15.1.4

Apply the power rule and multiply exponents,

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

$\frac{3}{2 \cdot 5^{2 (\frac{1}{2})}}$

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.

$\frac{3}{2 \cdot 5^{2 (\frac{1}{2})}}$

Step 1.15.1.5.2

Rewrite the expression.

$\frac{3}{2 \cdot 5^{1}}$

Step 1.15.1.6

Evaluate the exponent.

$\frac{3}{2 \cdot 5}$

Step 1.15.2

Multiply

$2$ by

$5$ .

$\frac{3}{10}$

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

$\frac{3}{10}$ and a given point

$(3, 5)$ 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

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

$y - (5) = \frac{3}{10} \cdot (x - (3))$

Step 2.2

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

$y - 5 = \frac{3}{10} \cdot (x - 3)$

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Simplify

$\frac{3}{10} \cdot (x - 3)$ .

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Step 2.3.1.1

Rewrite.

$y - 5 = 0 + 0 + \frac{3}{10} \cdot (x - 3)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 5 = \frac{3}{10} \cdot (x - 3)$

Step 2.3.1.3

Apply the distributive property.

$y - 5 = \frac{3}{10} x + \frac{3}{10} \cdot - 3$

Step 2.3.1.4

Combine

$\frac{3}{10}$ and

$x$ .

$y - 5 = \frac{3 x}{10} + \frac{3}{10} \cdot - 3$

Step 2.3.1.5

Multiply

$\frac{3}{10} \cdot - 3$ .

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Step 2.3.1.5.1

Combine

$\frac{3}{10}$ and

$- 3$ .

$y - 5 = \frac{3 x}{10} + \frac{3 \cdot - 3}{10}$

Step 2.3.1.5.2

Multiply

$3$ by

$- 3$ .

$y - 5 = \frac{3 x}{10} + \frac{- 9}{10}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y - 5 = \frac{3 x}{10} - \frac{9}{10}$

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

$5$ to both sides of the equation.

$y = \frac{3 x}{10} - \frac{9}{10} + 5$

Step 2.3.2.2

To write

$5$ as a fraction with a common denominator, multiply by

$\frac{10}{10}$ .

$y = \frac{3 x}{10} - \frac{9}{10} + 5 \cdot \frac{10}{10}$

Step 2.3.2.3

Combine

$5$ and

$\frac{10}{10}$ .

$y = \frac{3 x}{10} - \frac{9}{10} + \frac{5 \cdot 10}{10}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{3 x}{10} + \frac{- 9 + 5 \cdot 10}{10}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply

$5$ by

$10$ .

$y = \frac{3 x}{10} + \frac{- 9 + 50}{10}$

Step 2.3.2.5.2

Add

$- 9$ and

$50$ .

$y = \frac{3 x}{10} + \frac{41}{10}$

Step 2.3.3

Reorder terms.

$y = \frac{3}{10} x + \frac{41}{10}$

Step 3