Calculus Examples

$y = \sqrt{x}$ , $(9, 3)$

Step 1

Find the first derivative and evaluate at $x = 9$ and $y = 3$ to find the slope of the tangent line.

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Step 1.1

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 1.2

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 1.3

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 1.4

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

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.6.2

Subtract $2$ from $1$ .

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

Step 1.7

Move the negative in front of the fraction.

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

Step 1.8

Simplify.

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Step 1.8.1

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 1.8.2

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

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

Step 1.9

Evaluate the derivative at $x = 9$ .

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

Step 1.10

Simplify.

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Step 1.10.1

Simplify the denominator.

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Step 1.10.1.1

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

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

Step 1.10.1.2

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

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

Step 1.10.1.3

Cancel the common factor of $2$ .

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Step 1.10.1.3.1

Cancel the common factor.

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

Step 1.10.1.3.2

Rewrite the expression.

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

Step 1.10.1.4

Evaluate the exponent.

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

Step 1.10.2

Multiply $2$ by $3$ .

$\frac{1}{6}$

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 $\frac{1}{6}$ and a given point $(9, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = \frac{1}{6} \cdot (x - (9))$

Step 2.2

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

$y - 3 = \frac{1}{6} \cdot (x - 9)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{1}{6} \cdot (x - 9)$ .

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Step 2.3.1.1

Rewrite.

$y - 3 = 0 + 0 + \frac{1}{6} \cdot (x - 9)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 3 = \frac{1}{6} \cdot (x - 9)$

Step 2.3.1.3

Apply the distributive property.

$y - 3 = \frac{1}{6} x + \frac{1}{6} \cdot - 9$

Step 2.3.1.4

Combine $\frac{1}{6}$ and $x$ .

$y - 3 = \frac{x}{6} + \frac{1}{6} \cdot - 9$

Step 2.3.1.5

Cancel the common factor of $3$ .

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Step 2.3.1.5.1

Factor $3$ out of $6$ .

$y - 3 = \frac{x}{6} + \frac{1}{3 (2)} \cdot - 9$

Step 2.3.1.5.2

Factor $3$ out of $- 9$ .

$y - 3 = \frac{x}{6} + \frac{1}{3 \cdot 2} \cdot (3 \cdot - 3)$

Step 2.3.1.5.3

Cancel the common factor.

$y - 3 = \frac{x}{6} + \frac{1}{3 \cdot 2} \cdot (3 \cdot - 3)$

Step 2.3.1.5.4

Rewrite the expression.

$y - 3 = \frac{x}{6} + \frac{1}{2} \cdot - 3$

Step 2.3.1.6

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

$y - 3 = \frac{x}{6} + \frac{- 3}{2}$

Step 2.3.1.7

Move the negative in front of the fraction.

$y - 3 = \frac{x}{6} - \frac{3}{2}$

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 $3$ to both sides of the equation.

$y = \frac{x}{6} - \frac{3}{2} + 3$

Step 2.3.2.2

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

$y = \frac{x}{6} - \frac{3}{2} + 3 \cdot \frac{2}{2}$

Step 2.3.2.3

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

$y = \frac{x}{6} - \frac{3}{2} + \frac{3 \cdot 2}{2}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{x}{6} + \frac{- 3 + 3 \cdot 2}{2}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $3$ by $2$ .

$y = \frac{x}{6} + \frac{- 3 + 6}{2}$

Step 2.3.2.5.2

Add $- 3$ and $6$ .

$y = \frac{x}{6} + \frac{3}{2}$

Step 2.3.3

Reorder terms.

$y = \frac{1}{6} x + \frac{3}{2}$

Step 3