Calculus Examples

f (x) = x^{\frac{1}{2}} + 2 x^{- 1}

$f (x) = x^{\frac{1}{2}} + 2 x^{- 1}$ at the point

(4, \frac{5}{2})

$(4, \frac{5}{2})$

Step 1

Find the first derivative and evaluate at

x = 4

$x = 4$ and

y = \frac{5}{2}

$y = \frac{5}{2}$ to find the slope of the tangent line.

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Step 1.1

By the Sum Rule, the derivative of

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

$x^{\frac{1}{2}} + 2 x^{- 1}$ with respect to

x

$x$ is

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

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

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

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

Step 1.2

Evaluate

\frac{d}{d x} [x^{\frac{1}{2}}]

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

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Step 1.2.1

Differentiate using the Power Rule which states that

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

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

n x^{n - 1}

$n x^{n - 1}$ where

n = \frac{1}{2}

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

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

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

Step 1.2.2

To write

- 1

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

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 1.2.3

Combine

- 1

$- 1$ and

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 1.2.4

Combine the numerators over the common denominator.

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

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

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 1.2.5.2

Subtract

2

$2$ from

1

$1$ .

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

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

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

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

Step 1.2.6

Move the negative in front of the fraction.

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

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

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

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

Step 1.3

Evaluate

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

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

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Step 1.3.1

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

2 x^{- 1}

$2 x^{- 1}$ with respect to

x

$x$ is

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

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

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

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

Step 1.3.2

Differentiate using the Power Rule which states that

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

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

n x^{n - 1}

$n x^{n - 1}$ where

n = - 1

$n = - 1$ .

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

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

Step 1.3.3

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

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

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

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}}

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

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

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

Step 1.4.2

Rewrite the expression using the negative exponent rule

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

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

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

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

Step 1.4.3

Combine terms.

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Step 1.4.3.1

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

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

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

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

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

Step 1.4.3.2

Combine

- 2

$- 2$ and

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

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

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

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

Step 1.4.3.3

Move the negative in front of the fraction.

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

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

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

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

Step 1.4.4

Reorder terms.

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

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

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

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

Step 1.5

Evaluate the derivative at

x = 4

$x = 4$ .

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

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

Step 1.6

Simplify.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Raise

4

$4$ to the power of

2

$2$ .

- \frac{2}{16} + \frac{1}{2 {(4)}^{\frac{1}{2}}}

$- \frac{2}{16} + \frac{1}{2 {(4)}^{\frac{1}{2}}}$

Step 1.6.1.2

Cancel the common factor of

2

$2$ and

16

$16$ .

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Step 1.6.1.2.1

Factor

2

$2$ out of

2

$2$ .

- \frac{2 (1)}{16} + \frac{1}{2 {(4)}^{\frac{1}{2}}}

$- \frac{2 (1)}{16} + \frac{1}{2 {(4)}^{\frac{1}{2}}}$

Step 1.6.1.2.2

Cancel the common factors.

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Step 1.6.1.2.2.1

Factor

2

$2$ out of

16

$16$ .

- \frac{2 \cdot 1}{2 \cdot 8} + \frac{1}{2 {(4)}^{\frac{1}{2}}}

$- \frac{2 \cdot 1}{2 \cdot 8} + \frac{1}{2 {(4)}^{\frac{1}{2}}}$

Step 1.6.1.2.2.2

Cancel the common factor.

$- \frac{2 \cdot 1}{2 \cdot 8} + \frac{1}{2 {(4)}^{\frac{1}{2}}}$

Step 1.6.1.2.2.3

Rewrite the expression.

$- \frac{1}{8} + \frac{1}{2 {(4)}^{\frac{1}{2}}}$

Step 1.6.1.3

Simplify the denominator.

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Step 1.6.1.3.1

Rewrite

$4$ as

$2^{2}$ .

$- \frac{1}{8} + \frac{1}{2 \cdot {(2^{2})}^{\frac{1}{2}}}$

Step 1.6.1.3.2

Multiply the exponents in

${(2^{2})}^{\frac{1}{2}}$ .

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Step 1.6.1.3.2.1

Apply the power rule and multiply exponents,

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

$- \frac{1}{8} + \frac{1}{2 \cdot 2^{2 (\frac{1}{2})}}$

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.

$- \frac{1}{8} + \frac{1}{2 \cdot 2^{2 (\frac{1}{2})}}$

Step 1.6.1.3.2.2.2

Rewrite the expression.

$- \frac{1}{8} + \frac{1}{2 \cdot 2^{1}}$

Step 1.6.1.3.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{1}{8} + \frac{1}{2^{1 + 1}}$

Step 1.6.1.3.4

Add

$1$ and

$1$ .

$- \frac{1}{8} + \frac{1}{2^{2}}$

Step 1.6.1.4

Raise

$2$ to the power of

$2$ .

$- \frac{1}{8} + \frac{1}{4}$

Step 1.6.2

To write

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

$\frac{2}{2}$ .

$- \frac{1}{8} + \frac{1}{4} \cdot \frac{2}{2}$

Step 1.6.3

Write each expression with a common denominator of

$8$ , by multiplying each by an appropriate factor of

$1$ .

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Step 1.6.3.1

Multiply

$\frac{1}{4}$ by

$\frac{2}{2}$ .

$- \frac{1}{8} + \frac{2}{4 \cdot 2}$

Step 1.6.3.2

Multiply

$4$ by

$2$ .

$- \frac{1}{8} + \frac{2}{8}$

Step 1.6.4

Combine the numerators over the common denominator.

$\frac{- 1 + 2}{8}$

Step 1.6.5

Add

$- 1$ and

$2$ .

$\frac{1}{8}$

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

$\frac{1}{8}$ 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

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

$y - (\frac{5}{2}) = \frac{1}{8} \cdot (x - (4))$

Step 2.2

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

$y - \frac{5}{2} = \frac{1}{8} \cdot (x - 4)$

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Simplify

$\frac{1}{8} \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

$y - \frac{5}{2} = 0 + 0 + \frac{1}{8} \cdot (x - 4)$

Step 2.3.1.2

Simplify by adding zeros.

$y - \frac{5}{2} = \frac{1}{8} \cdot (x - 4)$

Step 2.3.1.3

Apply the distributive property.

$y - \frac{5}{2} = \frac{1}{8} x + \frac{1}{8} \cdot - 4$

Step 2.3.1.4

Combine

$\frac{1}{8}$ and

$x$ .

$y - \frac{5}{2} = \frac{x}{8} + \frac{1}{8} \cdot - 4$

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

$8$ .

$y - \frac{5}{2} = \frac{x}{8} + \frac{1}{4 (2)} \cdot - 4$

Step 2.3.1.5.2

Factor

$4$ out of

$- 4$ .

$y - \frac{5}{2} = \frac{x}{8} + \frac{1}{4 \cdot 2} \cdot (4 \cdot - 1)$

Step 2.3.1.5.3

Cancel the common factor.

$y - \frac{5}{2} = \frac{x}{8} + \frac{1}{4 \cdot 2} \cdot (4 \cdot - 1)$

Step 2.3.1.5.4

Rewrite the expression.

$y - \frac{5}{2} = \frac{x}{8} + \frac{1}{2} \cdot - 1$

Step 2.3.1.6

Combine

$\frac{1}{2}$ and

$- 1$ .

$y - \frac{5}{2} = \frac{x}{8} + \frac{- 1}{2}$

Step 2.3.1.7

Move the negative in front of the fraction.

$y - \frac{5}{2} = \frac{x}{8} - \frac{1}{2}$

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.

$y = \frac{x}{8} - \frac{1}{2} + \frac{5}{2}$

Step 2.3.2.2

Combine the numerators over the common denominator.

$y = \frac{x}{8} + \frac{- 1 + 5}{2}$

Step 2.3.2.3

Add

$- 1$ and

$5$ .

$y = \frac{x}{8} + \frac{4}{2}$

Step 2.3.2.4

Divide

$4$ by

$2$ .

$y = \frac{x}{8} + 2$

Step 2.3.3

Reorder terms.

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

Step 3