Calculus Examples

$\frac{2}{\sqrt{x}}$ , $x = 4$

Step 1

Write $\frac{2}{\sqrt{x}}$ as an equation.

$y = \frac{2}{\sqrt{x}}$

Step 2

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

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Step 2.1

Substitute $4$ in for $x$ .

$y = \frac{2}{\sqrt{4}}$

Step 2.2

Simplify $\frac{2}{\sqrt{4}}$ .

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Step 2.2.1

Remove parentheses.

$y = \frac{2}{\sqrt{4}}$

Step 2.2.2

Simplify the denominator.

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

$y = \frac{2}{2}$

Step 2.2.3

Divide $2$ by $2$ .

$y = 1$

Step 3

Find the first derivative and evaluate at $x = 4$ and $y = 1$ to find the slope of the tangent line.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Apply basic rules of exponents.

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Step 3.3.1

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

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

Step 3.3.2

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

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Step 3.3.2.1

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

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

Step 3.3.2.2

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

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

Step 3.3.2.3

Move the negative in front of the fraction.

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Combine the numerators over the common denominator.

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

Step 3.8

Simplify the numerator.

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Step 3.8.1

Multiply $- 1$ by $2$ .

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

Step 3.8.2

Subtract $2$ from $- 1$ .

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

Step 3.9

Move the negative in front of the fraction.

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

Step 3.10

Combine $x^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$2 (- \frac{x^{- \frac{3}{2}}}{2})$

Step 3.11

Multiply $- 1$ by $2$ .

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

Step 3.12

Combine $- 2$ and $\frac{x^{- \frac{3}{2}}}{2}$ .

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

Step 3.13

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

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

Step 3.14

Factor $2$ out of $- 2$ .

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

Step 3.15

Cancel the common factors.

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Step 3.15.1

Factor $2$ out of $2 x^{\frac{3}{2}}$ .

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

Step 3.15.2

Cancel the common factor.

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

Step 3.15.3

Rewrite the expression.

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

Step 3.16

Move the negative in front of the fraction.

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

Step 3.17

Evaluate the derivative at $x = 4$ .

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

Step 3.18

Simplify the denominator.

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Step 3.18.1

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

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

Step 3.18.2

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

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

Step 3.18.3

Cancel the common factor of $2$ .

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Step 3.18.3.1

Cancel the common factor.

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

Step 3.18.3.2

Rewrite the expression.

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

Step 3.18.4

Raise $2$ to the power of $3$ .

$- \frac{1}{8}$

Step 4

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 4.1

Use the slope $- \frac{1}{8}$ and a given point $(4, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 4.2

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

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

Step 4.3

Solve for $y$ .

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Step 4.3.1

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

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Step 4.3.1.1

Rewrite.

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

Step 4.3.1.2

Simplify by adding zeros.

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

Step 4.3.1.3

Apply the distributive property.

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

Step 4.3.1.4

Combine $x$ and $\frac{1}{8}$ .

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

Step 4.3.1.5

Cancel the common factor of $4$ .

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Step 4.3.1.5.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

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

Step 4.3.1.5.2

Factor $4$ out of $8$ .

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

Step 4.3.1.5.3

Factor $4$ out of $- 4$ .

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

Step 4.3.1.5.4

Cancel the common factor.

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

Step 4.3.1.5.5

Rewrite the expression.

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

Step 4.3.1.6

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

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

Step 4.3.1.7

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 4.3.2.1

Add $1$ to both sides of the equation.

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

Step 4.3.2.2

Write $1$ as a fraction with a common denominator.

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

Step 4.3.2.3

Combine the numerators over the common denominator.

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

Step 4.3.2.4

Add $1$ and $2$ .

$y = - \frac{x}{8} + \frac{3}{2}$

Step 4.3.3

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

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Step 4.3.3.1

Reorder terms.

$y = - (\frac{1}{8} x) + \frac{3}{2}$

Step 4.3.3.2

Remove parentheses.

$y = - \frac{1}{8} x + \frac{3}{2}$

Step 5