Calculus Examples

$y = x \sqrt{x}$ , $(9, 27)$

Step 1

Find the first derivative and evaluate at $x = 9$ and $y = 27$ 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 $x^{\frac{1}{2}}$ .

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

Step 1.2

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

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Step 1.2.1

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

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Step 1.2.1.1

Raise $x$ to the power of $1$ .

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

Step 1.2.1.2

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

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

Step 1.2.2

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

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

Step 1.2.3

Combine the numerators over the common denominator.

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

Step 1.2.4

Add $2$ and $1$ .

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

Step 1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.7.2

Subtract $2$ from $3$ .

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

Step 1.8

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

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

Step 1.9

Evaluate the derivative at $x = 9$ .

$\frac{3 {(9)}^{\frac{1}{2}}}{2}$

Step 1.10

Simplify.

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Step 1.10.1

Simplify the numerator.

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Step 1.10.1.1

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

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

Step 1.10.1.2

Multiply the exponents in ${(3^{2})}^{\frac{1}{2}}$ .

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Step 1.10.1.2.1

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

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

Step 1.10.1.2.2

Cancel the common factor of $2$ .

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Step 1.10.1.2.2.1

Cancel the common factor.

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

Step 1.10.1.2.2.2

Rewrite the expression.

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

Step 1.10.1.3

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

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

Step 1.10.1.4

Add $1$ and $1$ .

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

Step 1.10.2

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

$\frac{9}{2}$

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 $\frac{9}{2}$ and a given point $(9, 27)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (27) = \frac{9}{2} \cdot (x - (9))$

Step 2.2

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

$y - 27 = \frac{9}{2} \cdot (x - 9)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{9}{2} \cdot (x - 9)$ .

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Step 2.3.1.1

Rewrite.

$y - 27 = 0 + 0 + \frac{9}{2} \cdot (x - 9)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 27 = \frac{9}{2} \cdot (x - 9)$

Step 2.3.1.3

Apply the distributive property.

$y - 27 = \frac{9}{2} x + \frac{9}{2} \cdot - 9$

Step 2.3.1.4

Combine $\frac{9}{2}$ and $x$ .

$y - 27 = \frac{9 x}{2} + \frac{9}{2} \cdot - 9$

Step 2.3.1.5

Multiply $\frac{9}{2} \cdot - 9$ .

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Step 2.3.1.5.1

Combine $\frac{9}{2}$ and $- 9$ .

$y - 27 = \frac{9 x}{2} + \frac{9 \cdot - 9}{2}$

Step 2.3.1.5.2

Multiply $9$ by $- 9$ .

$y - 27 = \frac{9 x}{2} + \frac{- 81}{2}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y - 27 = \frac{9 x}{2} - \frac{81}{2}$

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 $27$ to both sides of the equation.

$y = \frac{9 x}{2} - \frac{81}{2} + 27$

Step 2.3.2.2

To write $27$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{9 x}{2} - \frac{81}{2} + 27 \cdot \frac{2}{2}$

Step 2.3.2.3

Combine $27$ and $\frac{2}{2}$ .

$y = \frac{9 x}{2} - \frac{81}{2} + \frac{27 \cdot 2}{2}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{9 x}{2} + \frac{- 81 + 27 \cdot 2}{2}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $27$ by $2$ .

$y = \frac{9 x}{2} + \frac{- 81 + 54}{2}$

Step 2.3.2.5.2

Add $- 81$ and $54$ .

$y = \frac{9 x}{2} + \frac{- 27}{2}$

Step 2.3.2.6

Move the negative in front of the fraction.

$y = \frac{9 x}{2} - \frac{27}{2}$

Step 2.3.3

Reorder terms.

$y = \frac{9}{2} x - \frac{27}{2}$

Step 3