Calculus Examples

$f (x) = \sqrt{x^{2} + 24}$ , $x = 5$

Step 1

Find the corresponding $y$ -value to $x = 5$ .

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Step 1.1

Substitute $5$ in for $x$ .

$y = \sqrt{{(5)}^{2} + 24}$

Step 1.2

Simplify $\sqrt{{(5)}^{2} + 24}$ .

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Step 1.2.1

Raise $5$ to the power of $2$ .

$y = \sqrt{25 + 24}$

Step 1.2.2

Add $25$ and $24$ .

$y = \sqrt{49}$

Step 1.2.3

Rewrite $49$ as $7^{2}$ .

$y = \sqrt{7^{2}}$

Step 1.2.4

Pull terms out from under the radical, assuming positive real numbers.

$y = 7$

Step 2

Find the first derivative and evaluate at $x = 5$ and $y = 7$ 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 $\sqrt{x^{2} + 24}$ as ${(x^{2} + 24)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(x^{2} + 24)}^{\frac{1}{2}}]$

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 $g (x) = x^{2} + 24$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 24$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{2} + 24]$

Step 2.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 24]$

Step 2.2.3

Replace all occurrences of $u$ with $x^{2} + 24$ .

$\frac{1}{2} {(x^{2} + 24)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 24]$

Step 2.3

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

$\frac{1}{2} {(x^{2} + 24)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{2} + 24]$

Step 2.4

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

$\frac{1}{2} {(x^{2} + 24)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 24]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(x^{2} + 24)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 24]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(x^{2} + 24)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{2} + 24]$

Step 2.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(x^{2} + 24)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{2} + 24]$

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(x^{2} + 24)}^{- \frac{1}{2}} \frac{d}{d x} [x^{2} + 24]$

Step 2.7.2

Combine $\frac{1}{2}$ and ${(x^{2} + 24)}^{- \frac{1}{2}}$ .

$\frac{{(x^{2} + 24)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{2} + 24]$

Step 2.7.3

Move ${(x^{2} + 24)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(x^{2} + 24)}^{\frac{1}{2}}} \frac{d}{d x} [x^{2} + 24]$

Step 2.8

By the Sum Rule, the derivative of $x^{2} + 24$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [24]$ .

$\frac{1}{2 {(x^{2} + 24)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [24])$

Step 2.9

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

$\frac{1}{2 {(x^{2} + 24)}^{\frac{1}{2}}} (2 x + \frac{d}{d x} [24])$

Step 2.10

Since $24$ is constant with respect to $x$ , the derivative of $24$ with respect to $x$ is $0$ .

$\frac{1}{2 {(x^{2} + 24)}^{\frac{1}{2}}} (2 x + 0)$

Step 2.11

Simplify terms.

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Step 2.11.1

Add $2 x$ and $0$ .

$\frac{1}{2 {(x^{2} + 24)}^{\frac{1}{2}}} (2 x)$

Step 2.11.2

Combine $2$ and $\frac{1}{2 {(x^{2} + 24)}^{\frac{1}{2}}}$ .

$\frac{2}{2 {(x^{2} + 24)}^{\frac{1}{2}}} x$

Step 2.11.3

Combine $\frac{2}{2 {(x^{2} + 24)}^{\frac{1}{2}}}$ and $x$ .

$\frac{2 x}{2 {(x^{2} + 24)}^{\frac{1}{2}}}$

Step 2.11.4

Cancel the common factor.

$\frac{2 x}{2 {(x^{2} + 24)}^{\frac{1}{2}}}$

Step 2.11.5

Rewrite the expression.

$\frac{x}{{(x^{2} + 24)}^{\frac{1}{2}}}$

Step 2.12

Evaluate the derivative at $x = 5$ .

$\frac{5}{{({(5)}^{2} + 24)}^{\frac{1}{2}}}$

Step 2.13

Simplify the denominator.

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Step 2.13.1

Raise $5$ to the power of $2$ .

$\frac{5}{{(25 + 24)}^{\frac{1}{2}}}$

Step 2.13.2

Add $25$ and $24$ .

$\frac{5}{49^{\frac{1}{2}}}$

Step 2.13.3

Rewrite $49$ as $7^{2}$ .

$\frac{5}{{(7^{2})}^{\frac{1}{2}}}$

Step 2.13.4

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

$\frac{5}{7^{2 (\frac{1}{2})}}$

Step 2.13.5

Cancel the common factor of $2$ .

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Step 2.13.5.1

Cancel the common factor.

$\frac{5}{7^{2 (\frac{1}{2})}}$

Step 2.13.5.2

Rewrite the expression.

$\frac{5}{7^{1}}$

Step 2.13.6

Evaluate the exponent.

$\frac{5}{7}$

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 $\frac{5}{7}$ and a given point $(5, 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (7) = \frac{5}{7} \cdot (x - (5))$

Step 3.2

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

$y - 7 = \frac{5}{7} \cdot (x - 5)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $\frac{5}{7} \cdot (x - 5)$ .

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Step 3.3.1.1

Rewrite.

$y - 7 = 0 + 0 + \frac{5}{7} \cdot (x - 5)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 7 = \frac{5}{7} \cdot (x - 5)$

Step 3.3.1.3

Apply the distributive property.

$y - 7 = \frac{5}{7} x + \frac{5}{7} \cdot - 5$

Step 3.3.1.4

Combine $\frac{5}{7}$ and $x$ .

$y - 7 = \frac{5 x}{7} + \frac{5}{7} \cdot - 5$

Step 3.3.1.5

Multiply $\frac{5}{7} \cdot - 5$ .

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Step 3.3.1.5.1

Combine $\frac{5}{7}$ and $- 5$ .

$y - 7 = \frac{5 x}{7} + \frac{5 \cdot - 5}{7}$

Step 3.3.1.5.2

Multiply $5$ by $- 5$ .

$y - 7 = \frac{5 x}{7} + \frac{- 25}{7}$

Step 3.3.1.6

Move the negative in front of the fraction.

$y - 7 = \frac{5 x}{7} - \frac{25}{7}$

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 $7$ to both sides of the equation.

$y = \frac{5 x}{7} - \frac{25}{7} + 7$

Step 3.3.2.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = \frac{5 x}{7} - \frac{25}{7} + 7 \cdot \frac{7}{7}$

Step 3.3.2.3

Combine $7$ and $\frac{7}{7}$ .

$y = \frac{5 x}{7} - \frac{25}{7} + \frac{7 \cdot 7}{7}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$y = \frac{5 x}{7} + \frac{- 25 + 7 \cdot 7}{7}$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $7$ by $7$ .

$y = \frac{5 x}{7} + \frac{- 25 + 49}{7}$

Step 3.3.2.5.2

Add $- 25$ and $49$ .

$y = \frac{5 x}{7} + \frac{24}{7}$

Step 3.3.3

Reorder terms.

$y = \frac{5}{7} x + \frac{24}{7}$

Step 4