Calculus Examples

$f (x) = \sqrt{x}$ , $x = \frac{4}{9}$

Step 1

Find the corresponding $y$ -value to $x = \frac{4}{9}$ .

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Step 1.1

Substitute $\frac{4}{9}$ in for $x$ .

$y = \sqrt{\frac{4}{9}}$

Step 1.2

Simplify $\sqrt{\frac{4}{9}}$ .

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Step 1.2.1

Remove parentheses.

$y = \sqrt{\frac{4}{9}}$

Step 1.2.2

Rewrite $\sqrt{\frac{4}{9}}$ as $\frac{\sqrt{4}}{\sqrt{9}}$ .

$y = \frac{\sqrt{4}}{\sqrt{9}}$

Step 1.2.3

Simplify the numerator.

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Step 1.2.3.1

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

$y = \frac{\sqrt{2^{2}}}{\sqrt{9}}$

Step 1.2.3.2

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

$y = \frac{2}{\sqrt{9}}$

Step 1.2.4

Simplify the denominator.

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Step 1.2.4.1

Rewrite $9$ as $3^{2}$ .

$y = \frac{2}{\sqrt{3^{2}}}$

Step 1.2.4.2

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

$y = \frac{2}{3}$

Step 2

Find the first derivative and evaluate at $x = \frac{4}{9}$ and $y = \frac{2}{3}$ to find the slope of the tangent line.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.6.2

Subtract $2$ from $1$ .

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

Step 2.7

Move the negative in front of the fraction.

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

Step 2.8

Simplify.

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Step 2.8.1

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

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

Step 2.8.2

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

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

Step 2.9

Evaluate the derivative at $x = \frac{4}{9}$ .

$\frac{1}{2 {(\frac{4}{9})}^{\frac{1}{2}}}$

Step 2.10

Simplify.

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Step 2.10.1

Simplify the denominator.

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Step 2.10.1.1

Apply the product rule to $\frac{4}{9}$ .

$\frac{1}{2 \frac{4^{\frac{1}{2}}}{9^{\frac{1}{2}}}}$

Step 2.10.1.2

Simplify the numerator.

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Step 2.10.1.2.1

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

$\frac{1}{2 \frac{{(2^{2})}^{\frac{1}{2}}}{9^{\frac{1}{2}}}}$

Step 2.10.1.2.2

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

$\frac{1}{2 \frac{2^{2 (\frac{1}{2})}}{9^{\frac{1}{2}}}}$

Step 2.10.1.2.3

Cancel the common factor of $2$ .

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Step 2.10.1.2.3.1

Cancel the common factor.

$\frac{1}{2 \frac{2^{2 (\frac{1}{2})}}{9^{\frac{1}{2}}}}$

Step 2.10.1.2.3.2

Rewrite the expression.

$\frac{1}{2 \frac{2^{1}}{9^{\frac{1}{2}}}}$

Step 2.10.1.2.4

Evaluate the exponent.

$\frac{1}{2 \frac{2}{9^{\frac{1}{2}}}}$

Step 2.10.1.3

Simplify the denominator.

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Step 2.10.1.3.1

Rewrite $9$ as $3^{2}$ .

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

Step 2.10.1.3.2

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

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

Step 2.10.1.3.3

Cancel the common factor of $2$ .

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Step 2.10.1.3.3.1

Cancel the common factor.

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

Step 2.10.1.3.3.2

Rewrite the expression.

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

Step 2.10.1.3.4

Evaluate the exponent.

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

Step 2.10.2

Combine fractions.

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Step 2.10.2.1

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

$\frac{1}{\frac{2 \cdot 2}{3}}$

Step 2.10.2.2

Multiply $2$ by $2$ .

$\frac{1}{\frac{4}{3}}$

Step 2.10.3

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{3}{4})$

Step 2.10.4

Multiply $\frac{3}{4}$ by $1$ .

$\frac{3}{4}$

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 $\frac{3}{4}$ and a given point $(\frac{4}{9}, \frac{2}{3})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{2}{3}) = \frac{3}{4} \cdot (x - (\frac{4}{9}))$

Step 3.2

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

$y - \frac{2}{3} = \frac{3}{4} \cdot (x - \frac{4}{9})$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $\frac{3}{4} \cdot (x - \frac{4}{9})$ .

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Step 3.3.1.1

Rewrite.

$y - \frac{2}{3} = 0 + 0 + \frac{3}{4} \cdot (x - \frac{4}{9})$

Step 3.3.1.2

Simplify by adding zeros.

$y - \frac{2}{3} = \frac{3}{4} \cdot (x - \frac{4}{9})$

Step 3.3.1.3

Apply the distributive property.

$y - \frac{2}{3} = \frac{3}{4} x + \frac{3}{4} (- \frac{4}{9})$

Step 3.3.1.4

Combine $\frac{3}{4}$ and $x$ .

$y - \frac{2}{3} = \frac{3 x}{4} + \frac{3}{4} (- \frac{4}{9})$

Step 3.3.1.5

Cancel the common factor of $3$ .

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Step 3.3.1.5.1

Move the leading negative in $- \frac{4}{9}$ into the numerator.

$y - \frac{2}{3} = \frac{3 x}{4} + \frac{3}{4} \cdot \frac{- 4}{9}$

Step 3.3.1.5.2

Factor $3$ out of $9$ .

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

Step 3.3.1.5.3

Cancel the common factor.

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

Step 3.3.1.5.4

Rewrite the expression.

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

Step 3.3.1.6

Cancel the common factor of $4$ .

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Step 3.3.1.6.1

Factor $4$ out of $- 4$ .

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

Step 3.3.1.6.2

Cancel the common factor.

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

Step 3.3.1.6.3

Rewrite the expression.

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

Step 3.3.1.7

Move the negative in front of the fraction.

$y - \frac{2}{3} = \frac{3 x}{4} - \frac{1}{3}$

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 $\frac{2}{3}$ to both sides of the equation.

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

Step 3.3.2.2

Combine the numerators over the common denominator.

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

Step 3.3.2.3

Add $- 1$ and $2$ .

$y = \frac{3 x}{4} + \frac{1}{3}$

Step 3.3.3

Reorder terms.

$y = \frac{3}{4} x + \frac{1}{3}$

Step 4