Calculus Examples

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

Step 1

Find the corresponding $y$ -value to $x = 1$ .

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Step 1.1

Substitute $1$ in for $x$ .

$y = \sqrt{1} + \frac{2}{\sqrt{1}}$

Step 1.2

Simplify $\sqrt{1} + \frac{2}{\sqrt{1}}$ .

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Step 1.2.1

Remove parentheses.

$y = \sqrt{1} + \frac{2}{\sqrt{1}}$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Any root of $1$ is $1$ .

$y = 1 + \frac{2}{\sqrt{1}}$

Step 1.2.2.2

Any root of $1$ is $1$ .

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

Step 1.2.2.3

Divide $2$ by $1$ .

$y = 1 + 2$

Step 1.2.3

Add $1$ and $2$ .

$y = 3$

Step 2

Find the first derivative and evaluate at $x = 1$ and $y = 3$ to find the slope of the tangent line.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Combine the numerators over the common denominator.

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

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.2.6.2

Subtract $2$ from $1$ .

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

Step 2.2.7

Move the negative in front of the fraction.

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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 2.3.4.1

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

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

Step 2.3.4.2

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

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

Step 2.3.4.3

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

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

Step 2.3.5

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

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

Step 2.3.6

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

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Step 2.3.6.1

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

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

Step 2.3.6.2

Cancel the common factor of $2$ .

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Step 2.3.6.2.1

Factor $2$ out of $- 2$ .

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

Step 2.3.6.2.2

Cancel the common factor.

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

Step 2.3.6.2.3

Rewrite the expression.

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

Step 2.3.7

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

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

Step 2.3.8

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

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

Step 2.3.9

Combine the numerators over the common denominator.

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

Step 2.3.10

Simplify the numerator.

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Step 2.3.10.1

Multiply $- 1$ by $2$ .

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

Step 2.3.10.2

Subtract $2$ from $1$ .

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

Step 2.3.11

Move the negative in front of the fraction.

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

Step 2.3.12

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

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

Step 2.3.13

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

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

Step 2.3.14

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

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Step 2.3.14.1

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

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

Step 2.3.14.2

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

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

Step 2.3.14.3

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

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

Step 2.3.14.4

Combine the numerators over the common denominator.

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

Step 2.3.14.5

Simplify the numerator.

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Step 2.3.14.5.1

Multiply $- 1$ by $2$ .

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

Step 2.3.14.5.2

Subtract $2$ from $- 1$ .

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

Step 2.3.14.6

Move the negative in front of the fraction.

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

Step 2.3.15

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

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

Step 2.3.16

Multiply $- 1$ by $2$ .

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

Step 2.3.17

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

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

Step 2.3.18

Factor $2$ out of $- 2$ .

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

Step 2.3.19

Cancel the common factors.

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Step 2.3.19.1

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

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

Step 2.3.19.2

Cancel the common factor.

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

Step 2.3.19.3

Rewrite the expression.

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

Step 2.3.20

Move the negative in front of the fraction.

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

Step 2.4

Simplify.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.5

Evaluate the derivative at $x = 1$ .

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

Step 2.6

Simplify.

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Step 2.6.1

Simplify each term.

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Step 2.6.1.1

One to any power is one.

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

Step 2.6.1.2

Multiply $2$ by $1$ .

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

Step 2.6.1.3

One to any power is one.

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

Step 2.6.1.4

Cancel the common factor of $1$ .

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Step 2.6.1.4.1

Cancel the common factor.

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

Step 2.6.1.4.2

Rewrite the expression.

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

Step 2.6.1.5

Multiply $- 1$ by $1$ .

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Combine the numerators over the common denominator.

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

Step 2.6.5

Simplify the numerator.

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Step 2.6.5.1

Multiply $- 1$ by $2$ .

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

Step 2.6.5.2

Subtract $2$ from $1$ .

$\frac{- 1}{2}$

Step 2.6.6

Move the negative in front of the fraction.

$- \frac{1}{2}$

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

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

Step 3.2

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

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

Step 3.3

Solve for $y$ .

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Step 3.3.1

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

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Step 3.3.1.1

Rewrite.

$y - 3 = 0 + 0 - \frac{1}{2} \cdot (x - 1)$

Step 3.3.1.2

Simplify by adding zeros.

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

Step 3.3.1.3

Apply the distributive property.

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

Step 3.3.1.4

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

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

Step 3.3.1.5

Multiply $- \frac{1}{2} \cdot - 1$ .

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Step 3.3.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.1.5.2

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

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

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 $3$ to both sides of the equation.

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

Step 3.3.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.3.2.3

Combine $3$ and $\frac{2}{2}$ .

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

Step 3.3.2.4

Combine the numerators over the common denominator.

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

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $3$ by $2$ .

$y = - \frac{x}{2} + \frac{1 + 6}{2}$

Step 3.3.2.5.2

Add $1$ and $6$ .

$y = - \frac{x}{2} + \frac{7}{2}$

Step 3.3.3

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

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Step 3.3.3.1

Reorder terms.

$y = - (\frac{1}{2} x) + \frac{7}{2}$

Step 3.3.3.2

Remove parentheses.

$y = - \frac{1}{2} x + \frac{7}{2}$

Step 4