Calculus Examples

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

Step 1

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

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Step 1.1

Simplify with factoring out.

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(3 x)}^{\frac{1}{2}}]$

Step 1.1.2

Factor $3$ out of $3 x$ .

$\frac{d}{d x} [{(3 (x))}^{\frac{1}{2}}]$

Step 1.1.3

Apply the product rule to $3 (x)$ .

$\frac{d}{d x} [3^{\frac{1}{2}} x^{\frac{1}{2}}]$

Step 1.2

Since $3^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $3^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $3^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$3^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.7.2

Subtract $2$ from $1$ .

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

Step 1.8

Move the negative in front of the fraction.

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

Evaluate the derivative at $x = 3$ .

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

Step 1.13

Simplify.

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Step 1.13.1

Cancel the common factor.

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

Step 1.13.2

Rewrite the expression.

$\frac{1}{2}$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

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

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

Step 2.3.1.5

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

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

Step 2.3.1.6

Move the negative in front of the fraction.

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

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.

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.2.4

Combine the numerators over the common denominator.

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

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $3$ by $2$ .

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

Step 2.3.2.5.2

Add $- 3$ and $6$ .

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

Step 2.3.3

Reorder terms.

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

Step 3