Calculus Examples

$f (x) = x^{\frac{1}{2}} + 2 x^{- 1}$ at the point $(4, \frac{5}{2})$

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 $x^{\frac{1}{2}} + 2 x^{- 1}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [2 x^{- 1}]$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Combine the numerators over the common denominator.

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

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply $- 1$ by $2$ .

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

Step 1.2.5.2

Subtract $2$ from $1$ .

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

Step 1.2.6

Move the negative in front of the fraction.

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

Step 1.3

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

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Step 1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{- 1}$ with respect to $x$ is $2 \frac{d}{d x} [x^{- 1}]$ .

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $- 1$ by $2$ .

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

Step 1.4

Simplify.

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Combine terms.

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Step 1.4.3.1

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

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

Step 1.4.3.2

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

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

Step 1.4.3.3

Move the negative in front of the fraction.

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

Step 1.4.4

Reorder terms.

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

Step 1.5

Evaluate the derivative at $x = 4$ .

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

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 $4$ to the power of $2$ .

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

Step 1.6.1.2

Cancel the common factor of $2$ and $16$ .

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Step 1.6.1.2.1

Factor $2$ out of $2$ .

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

Step 1.6.1.2.2

Cancel the common factors.

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Step 1.6.1.2.2.1

Factor $2$ out of $16$ .

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

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