Calculus Examples

$f (x) = \sqrt{x} + \frac{1}{\sqrt{x}}$ ; $(4, \frac{5}{2})$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = \frac{5}{2}$ to find the slope of the tangent line.

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Step 1.1

By the Sum Rule, the derivative of $\sqrt{x} + \frac{1}{\sqrt{x}}$ with respect to $x$ is $\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [\frac{1}{\sqrt{x}}]$ .

$\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [\frac{1}{\sqrt{x}}]$

Step 1.2

Evaluate $\frac{d}{d x} [\sqrt{x}]$ .

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

$\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [\frac{1}{\sqrt{x}}]$

Step 1.2.4

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

$\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [\frac{1}{\sqrt{x}}]$

Step 1.2.5

Combine the numerators over the common denominator.

$\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} [\frac{1}{\sqrt{x}}]$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $2$ .

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

Step 1.2.6.2

Subtract $2$ from $1$ .

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

Step 1.2.7

Move the negative in front of the fraction.

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

Step 1.3

Evaluate $\frac{d}{d x} [\frac{1}{\sqrt{x}}]$ .

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Step 1.3.1

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

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

Step 1.3.2

Rewrite $\frac{1}{x^{\frac{1}{2}}}$ as ${(x^{\frac{1}{2}})}^{- 1}$ .

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

Step 1.3.3

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

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Step 1.3.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{2}}$ .

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

Step 1.3.3.2

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

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

Step 1.3.3.3

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

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

Step 1.3.4

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

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

Step 1.3.5

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 2}$ .

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Step 1.3.5.1

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

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

Step 1.3.5.2

Cancel the common factor of $2$ .

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Step 1.3.5.2.1

Factor $2$ out of $- 2$ .

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

Step 1.3.5.2.2

Cancel the common factor.

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

Step 1.3.5.2.3

Rewrite the expression.

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

Step 1.3.6

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

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

Step 1.3.7

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

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

Step 1.3.8

Combine the numerators over the common denominator.

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

Step 1.3.9

Simplify the numerator.

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Step 1.3.9.1

Multiply $- 1$ by $2$ .

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

Step 1.3.9.2

Subtract $2$ from $1$ .

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

Step 1.3.10

Move the negative in front of the fraction.

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

Step 1.3.11

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

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

Step 1.3.12

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

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

Step 1.3.13

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

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Step 1.3.13.1

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

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

Step 1.3.13.2

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

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

Step 1.3.13.3

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

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

Step 1.3.13.4

Combine the numerators over the common denominator.

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

Step 1.3.13.5

Simplify the numerator.

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Step 1.3.13.5.1

Multiply $- 1$ by $2$ .

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

Step 1.3.13.5.2

Subtract $2$ from $- 1$ .

$\frac{1}{2} x^{- \frac{1}{2}} - \frac{x^{\frac{- 3}{2}}}{2}$

Step 1.3.13.6

Move the negative in front of the fraction.

$\frac{1}{2} x^{- \frac{1}{2}} - \frac{x^{- \frac{3}{2}}}{2}$

Step 1.3.14

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} x^{- \frac{1}{2}} - \frac{1}{2 x^{\frac{3}{2}}}$

Step 1.4

Simplify.

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Step 1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.4.2

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

$\frac{1}{2 x^{\frac{1}{2}}} - \frac{1}{2 x^{\frac{3}{2}}}$

Step 1.5

Evaluate the derivative at $x = 4$ .

$\frac{1}{2 {(4)}^{\frac{1}{2}}} - \frac{1}{2 {(4)}^{\frac{3}{2}}}$

Step 1.6

Simplify.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Simplify the denominator.

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Step 1.6.1.1.1

Rewrite $4$ as $2^{2}$ .

$\frac{1}{2 \cdot {(2^{2})}^{\frac{1}{2}}} - \frac{1}{2 {(4)}^{\frac{3}{2}}}$

Step 1.6.1.1.2

Multiply the exponents in ${(2^{2})}^{\frac{1}{2}}$ .

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Step 1.6.1.1.2.1

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

$\frac{1}{2 \cdot 2^{2 (\frac{1}{2})}} - \frac{1}{2 {(4)}^{\frac{3}{2}}}$

Step 1.6.1.1.2.2

Cancel the common factor of $2$ .

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Step 1.6.1.1.2.2.1

Cancel the common factor.

$\frac{1}{2 \cdot 2^{2 (\frac{1}{2})}} - \frac{1}{2 {(4)}^{\frac{3}{2}}}$

Step 1.6.1.1.2.2.2

Rewrite the expression.

$\frac{1}{2 \cdot 2^{1}} - \frac{1}{2 {(4)}^{\frac{3}{2}}}$

Step 1.6.1.1.3

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

$\frac{1}{2^{1 + 1}} - \frac{1}{2 {(4)}^{\frac{3}{2}}}$

Step 1.6.1.1.4

Add $1$ and $1$ .

$\frac{1}{2^{2}} - \frac{1}{2 {(4)}^{\frac{3}{2}}}$

Step 1.6.1.2

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

$\frac{1}{4} - \frac{1}{2 {(4)}^{\frac{3}{2}}}$

Step 1.6.1.3

Simplify the denominator.

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Step 1.6.1.3.1

Rewrite $4$ as $2^{2}$ .

$\frac{1}{4} - \frac{1}{2 \cdot {(2^{2})}^{\frac{3}{2}}}$

Step 1.6.1.3.2

Multiply the exponents in ${(2^{2})}^{\frac{3}{2}}$ .

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Step 1.6.1.3.2.1

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

$\frac{1}{4} - \frac{1}{2 \cdot 2^{2 (\frac{3}{2})}}$

Step 1.6.1.3.2.2

Cancel the common factor of $2$ .

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Step 1.6.1.3.2.2.1

Cancel the common factor.

$\frac{1}{4} - \frac{1}{2 \cdot 2^{2 (\frac{3}{2})}}$

Step 1.6.1.3.2.2.2

Rewrite the expression.

$\frac{1}{4} - \frac{1}{2 \cdot 2^{3}}$

Step 1.6.1.3.3

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

$\frac{1}{4} - \frac{1}{2^{1 + 3}}$

Step 1.6.1.3.4

Add $1$ and $3$ .

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

Step 1.6.1.4

Raise $2$ to the power of $4$ .

$\frac{1}{4} - \frac{1}{16}$

Step 1.6.2

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{4} \cdot \frac{4}{4} - \frac{1}{16}$

Step 1.6.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.6.3.1

Multiply $\frac{1}{4}$ by $\frac{4}{4}$ .

$\frac{4}{4 \cdot 4} - \frac{1}{16}$

Step 1.6.3.2

Multiply $4$ by $4$ .

$\frac{4}{16} - \frac{1}{16}$

Step 1.6.4

Combine the numerators over the common denominator.

$\frac{4 - 1}{16}$

Step 1.6.5

Subtract $1$ from $4$ .

$\frac{3}{16}$

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 $\frac{3}{16}$ and a given point $(4, \frac{5}{2})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{5}{2}) = \frac{3}{16} \cdot (x - (4))$

Step 2.2

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

$y - \frac{5}{2} = \frac{3}{16} \cdot (x - 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{3}{16} \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

$y - \frac{5}{2} = 0 + 0 + \frac{3}{16} \cdot (x - 4)$

Step 2.3.1.2

Simplify by adding zeros.

$y - \frac{5}{2} = \frac{3}{16} \cdot (x - 4)$

Step 2.3.1.3

Apply the distributive property.

$y - \frac{5}{2} = \frac{3}{16} x + \frac{3}{16} \cdot - 4$

Step 2.3.1.4

Combine $\frac{3}{16}$ and $x$ .

$y - \frac{5}{2} = \frac{3 x}{16} + \frac{3}{16} \cdot - 4$

Step 2.3.1.5

Cancel the common factor of $4$ .

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Step 2.3.1.5.1

Factor $4$ out of $16$ .

$y - \frac{5}{2} = \frac{3 x}{16} + \frac{3}{4 (4)} \cdot - 4$

Step 2.3.1.5.2

Factor $4$ out of $- 4$ .

$y - \frac{5}{2} = \frac{3 x}{16} + \frac{3}{4 \cdot 4} \cdot (4 \cdot - 1)$

Step 2.3.1.5.3

Cancel the common factor.

$y - \frac{5}{2} = \frac{3 x}{16} + \frac{3}{4 \cdot 4} \cdot (4 \cdot - 1)$

Step 2.3.1.5.4

Rewrite the expression.

$y - \frac{5}{2} = \frac{3 x}{16} + \frac{3}{4} \cdot - 1$

Step 2.3.1.6

Combine $\frac{3}{4}$ and $- 1$ .

$y - \frac{5}{2} = \frac{3 x}{16} + \frac{3 \cdot - 1}{4}$

Step 2.3.1.7

Simplify the expression.

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Step 2.3.1.7.1

Multiply $3$ by $- 1$ .

$y - \frac{5}{2} = \frac{3 x}{16} + \frac{- 3}{4}$

Step 2.3.1.7.2

Move the negative in front of the fraction.

$y - \frac{5}{2} = \frac{3 x}{16} - \frac{3}{4}$

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 $\frac{5}{2}$ to both sides of the equation.

$y = \frac{3 x}{16} - \frac{3}{4} + \frac{5}{2}$

Step 2.3.2.2

To write $\frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{3 x}{16} - \frac{3}{4} + \frac{5}{2} \cdot \frac{2}{2}$

Step 2.3.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.2.3.1

Multiply $\frac{5}{2}$ by $\frac{2}{2}$ .

$y = \frac{3 x}{16} - \frac{3}{4} + \frac{5 \cdot 2}{2 \cdot 2}$

Step 2.3.2.3.2

Multiply $2$ by $2$ .

$y = \frac{3 x}{16} - \frac{3}{4} + \frac{5 \cdot 2}{4}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{3 x}{16} + \frac{- 3 + 5 \cdot 2}{4}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $5$ by $2$ .

$y = \frac{3 x}{16} + \frac{- 3 + 10}{4}$

Step 2.3.2.5.2

Add $- 3$ and $10$ .

$y = \frac{3 x}{16} + \frac{7}{4}$

Step 2.3.3

Reorder terms.

$y = \frac{3}{16} x + \frac{7}{4}$

Step 3