Calculus Examples

$f (x) = \sqrt{4 x + 7}$ , $x = 4$

Step 1

Find the corresponding $y$ -value to $x = 4$ .

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Step 1.1

Substitute $4$ in for $x$ .

$y = \sqrt{4 (4) + 7}$

Step 1.2

Simplify $\sqrt{4 (4) + 7}$ .

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Step 1.2.1

Multiply $4$ by $4$ .

$y = \sqrt{16 + 7}$

Step 1.2.2

Add $16$ and $7$ .

$y = \sqrt{23}$

Step 2

Find the first derivative and evaluate at $x = 4$ and $y = \sqrt{23}$ to find the slope of the tangent line.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x + 7}$ as ${(4 x + 7)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(4 x + 7)}^{\frac{1}{2}}]$

Step 2.2

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 $g (x) = 4 x + 7$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $4 x + 7$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [4 x + 7]$

Step 2.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [4 x + 7]$

Step 2.2.3

Replace all occurrences of $u$ with $4 x + 7$ .

$\frac{1}{2} {(4 x + 7)}^{\frac{1}{2} - 1} \frac{d}{d x} [4 x + 7]$

Step 2.3

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

$\frac{1}{2} {(4 x + 7)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [4 x + 7]$

Step 2.4

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

$\frac{1}{2} {(4 x + 7)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [4 x + 7]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(4 x + 7)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [4 x + 7]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(4 x + 7)}^{\frac{1 - 2}{2}} \frac{d}{d x} [4 x + 7]$

Step 2.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(4 x + 7)}^{\frac{- 1}{2}} \frac{d}{d x} [4 x + 7]$

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(4 x + 7)}^{- \frac{1}{2}} \frac{d}{d x} [4 x + 7]$

Step 2.7.2

Combine $\frac{1}{2}$ and ${(4 x + 7)}^{- \frac{1}{2}}$ .

$\frac{{(4 x + 7)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [4 x + 7]$

Step 2.7.3

Move ${(4 x + 7)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(4 x + 7)}^{\frac{1}{2}}} \frac{d}{d x} [4 x + 7]$

Step 2.8

By the Sum Rule, the derivative of $4 x + 7$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [7]$ .

$\frac{1}{2 {(4 x + 7)}^{\frac{1}{2}}} (\frac{d}{d x} [4 x] + \frac{d}{d x} [7])$

Step 2.9

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

$\frac{1}{2 {(4 x + 7)}^{\frac{1}{2}}} (4 \frac{d}{d x} [x] + \frac{d}{d x} [7])$

Step 2.10

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

$\frac{1}{2 {(4 x + 7)}^{\frac{1}{2}}} (4 \cdot 1 + \frac{d}{d x} [7])$

Step 2.11

Multiply $4$ by $1$ .

$\frac{1}{2 {(4 x + 7)}^{\frac{1}{2}}} (4 + \frac{d}{d x} [7])$

Step 2.12

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{1}{2 {(4 x + 7)}^{\frac{1}{2}}} (4 + 0)$

Step 2.13

Simplify terms.

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Step 2.13.1

Add $4$ and $0$ .

$\frac{1}{2 {(4 x + 7)}^{\frac{1}{2}}} \cdot 4$

Step 2.13.2

Combine $\frac{1}{2 {(4 x + 7)}^{\frac{1}{2}}}$ and $4$ .

$\frac{4}{2 {(4 x + 7)}^{\frac{1}{2}}}$

Step 2.13.3

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2 {(4 x + 7)}^{\frac{1}{2}}}$

Step 2.14

Cancel the common factors.

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Step 2.14.1

Factor $2$ out of $2 {(4 x + 7)}^{\frac{1}{2}}$ .

$\frac{2 \cdot 2}{2 ({(4 x + 7)}^{\frac{1}{2}})}$

Step 2.14.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 {(4 x + 7)}^{\frac{1}{2}}}$

Step 2.14.3

Rewrite the expression.

$\frac{2}{{(4 x + 7)}^{\frac{1}{2}}}$

Step 2.15

Evaluate the derivative at $x = 4$ .

$\frac{2}{{(4 (4) + 7)}^{\frac{1}{2}}}$

Step 2.16

Simplify the denominator.

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Step 2.16.1

Multiply $4$ by $4$ .

$\frac{2}{{(16 + 7)}^{\frac{1}{2}}}$

Step 2.16.2

Add $16$ and $7$ .

$\frac{2}{23^{\frac{1}{2}}}$

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 $\frac{2}{23^{\frac{1}{2}}}$ and a given point $(4, \sqrt{23})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\sqrt{23}) = \frac{2}{23^{\frac{1}{2}}} \cdot (x - (4))$

Step 3.2

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

$y - \sqrt{23} = \frac{2}{23^{\frac{1}{2}}} \cdot (x - 4)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $\frac{2}{23^{\frac{1}{2}}} \cdot (x - 4)$ .

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Step 3.3.1.1

Rewrite.

$y - \sqrt{23} = 0 + 0 + \frac{2}{23^{\frac{1}{2}}} \cdot (x - 4)$

Step 3.3.1.2

Simplify by adding zeros.

$y - \sqrt{23} = \frac{2}{23^{\frac{1}{2}}} \cdot (x - 4)$

Step 3.3.1.3

Apply the distributive property.

$y - \sqrt{23} = \frac{2}{23^{\frac{1}{2}}} x + \frac{2}{23^{\frac{1}{2}}} \cdot - 4$

Step 3.3.1.4

Combine $\frac{2}{23^{\frac{1}{2}}}$ and $x$ .

$y - \sqrt{23} = \frac{2 x}{23^{\frac{1}{2}}} + \frac{2}{23^{\frac{1}{2}}} \cdot - 4$

Step 3.3.1.5

Multiply $\frac{2}{23^{\frac{1}{2}}} \cdot - 4$ .

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Step 3.3.1.5.1

Combine $\frac{2}{23^{\frac{1}{2}}}$ and $- 4$ .

$y - \sqrt{23} = \frac{2 x}{23^{\frac{1}{2}}} + \frac{2 \cdot - 4}{23^{\frac{1}{2}}}$

Step 3.3.1.5.2

Multiply $2$ by $- 4$ .

$y - \sqrt{23} = \frac{2 x}{23^{\frac{1}{2}}} + \frac{- 8}{23^{\frac{1}{2}}}$

Step 3.3.1.6

Move the negative in front of the fraction.

$y - \sqrt{23} = \frac{2 x}{23^{\frac{1}{2}}} - \frac{8}{23^{\frac{1}{2}}}$

Step 3.3.2

Add $\sqrt{23}$ to both sides of the equation.

$y = \frac{2 x}{23^{\frac{1}{2}}} - \frac{8}{23^{\frac{1}{2}}} + \sqrt{23}$

Step 3.3.3

Write in $y = m x + b$ form.

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Step 3.3.3.1

To write $\sqrt{23}$ as a fraction with a common denominator, multiply by $\frac{23^{\frac{1}{2}}}{23^{\frac{1}{2}}}$ .

$y = \frac{2 x}{23^{\frac{1}{2}}} - \frac{8}{23^{\frac{1}{2}}} + \sqrt{23} \cdot \frac{23^{\frac{1}{2}}}{23^{\frac{1}{2}}}$

Step 3.3.3.2

Combine $\sqrt{23}$ and $\frac{23^{\frac{1}{2}}}{23^{\frac{1}{2}}}$ .

$y = \frac{2 x}{23^{\frac{1}{2}}} - \frac{8}{23^{\frac{1}{2}}} + \frac{\sqrt{23} \cdot 23^{\frac{1}{2}}}{23^{\frac{1}{2}}}$

Step 3.3.3.3

Combine the numerators over the common denominator.

$y = \frac{2 x}{23^{\frac{1}{2}}} + \frac{- 8 + \sqrt{23} \cdot 23^{\frac{1}{2}}}{23^{\frac{1}{2}}}$

Step 3.3.3.4

Simplify the numerator.

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Step 3.3.3.4.1

Multiply $\sqrt{23} \cdot 23^{\frac{1}{2}}$ .

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Step 3.3.3.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{23}$ as $23^{\frac{1}{2}}$ .

$y = \frac{2 x}{23^{\frac{1}{2}}} + \frac{- 8 + 23^{\frac{1}{2}} \cdot 23^{\frac{1}{2}}}{23^{\frac{1}{2}}}$

Step 3.3.3.4.1.2

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

$y = \frac{2 x}{23^{\frac{1}{2}}} + \frac{- 8 + 23^{\frac{1}{2} + \frac{1}{2}}}{23^{\frac{1}{2}}}$

Step 3.3.3.4.1.3

Combine the numerators over the common denominator.

$y = \frac{2 x}{23^{\frac{1}{2}}} + \frac{- 8 + 23^{\frac{1 + 1}{2}}}{23^{\frac{1}{2}}}$

Step 3.3.3.4.1.4

Add $1$ and $1$ .

$y = \frac{2 x}{23^{\frac{1}{2}}} + \frac{- 8 + 23^{\frac{2}{2}}}{23^{\frac{1}{2}}}$

Step 3.3.3.4.1.5

Cancel the common factor of $2$ .

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Step 3.3.3.4.1.5.1

Cancel the common factor.

$y = \frac{2 x}{23^{\frac{1}{2}}} + \frac{- 8 + 23^{\frac{2}{2}}}{23^{\frac{1}{2}}}$

Step 3.3.3.4.1.5.2

Rewrite the expression.

$y = \frac{2 x}{23^{\frac{1}{2}}} + \frac{- 8 + 23^{1}}{23^{\frac{1}{2}}}$

Step 3.3.3.4.2

Evaluate the exponent.

$y = \frac{2 x}{23^{\frac{1}{2}}} + \frac{- 8 + 23}{23^{\frac{1}{2}}}$

Step 3.3.3.4.3

Add $- 8$ and $23$ .

$y = \frac{2 x}{23^{\frac{1}{2}}} + \frac{15}{23^{\frac{1}{2}}}$

Step 3.3.3.5

Reorder terms.

$y = \frac{2}{23^{\frac{1}{2}}} x + \frac{15}{23^{\frac{1}{2}}}$

Step 4