Find the Tangent Line at the Point y = square root of x ; (81,9)

$y = \sqrt{x}$ ; $(81, 9)$

Find $\frac{d y}{d x}$ and evaluate at $x = 81$ and $y = 9$ to find the slope of the tangent line at $x = 81$ and $y = 9$ .

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

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

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{\frac{1}{2}})$

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Differentiate the right side of the equation.

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

Reform the equation by setting the left side equal to the right side.

$y' = \frac{1}{2 x^{\frac{1}{2}}}$

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{1}{2 x^{\frac{1}{2}}}$

Evaluate at $x = 81$ and $y = 9$ .

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Replace the variable $x$ with $81$ in the expression.

$\frac{1}{2 {(81)}^{\frac{1}{2}}}$

Simplify the denominator.

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Rewrite $81$ as $9^{2}$ .

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $2$ by $9$ .

$\frac{1}{18}$

Plug in the slope of the tangent line and the $x$ and $y$ values of the point into the point–slope formula $y - y_{1} = m (x - x_{1})$ .

$y - (9) = \frac{1}{18} \cdot (x - (81))$

Simplify.

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The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Rewrite in slope-intercept form.

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Multiply $- 1$ by $9$ .

$y - 9 = \frac{1}{18} \cdot (x - (81))$

Simplify $\frac{1}{18} \cdot (x - (81))$ .

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Multiply $- 1$ by $81$ .

$y - 9 = \frac{1}{18} \cdot (x - 81)$

Apply the distributive property.

$y - 9 = \frac{1}{18} x + \frac{1}{18} \cdot - 81$

Combine $\frac{1}{18}$ and $x$ .

$y - 9 = \frac{x}{18} + \frac{1}{18} \cdot - 81$

Cancel the common factor of $9$ .

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Factor $9$ out of $18$ .

$y - 9 = \frac{x}{18} + \frac{1}{9 (2)} \cdot - 81$

Factor $9$ out of $- 81$ .

$y - 9 = \frac{x}{18} + \frac{1}{9 \cdot 2} \cdot (9 \cdot - 9)$

Cancel the common factor.

$y - 9 = \frac{x}{18} + \frac{1}{9 \cdot 2} \cdot (9 \cdot - 9)$

Rewrite the expression.

$y - 9 = \frac{x}{18} + \frac{1}{2} \cdot - 9$

Combine $\frac{1}{2}$ and $- 9$ .

$y - 9 = \frac{x}{18} + \frac{- 9}{2}$

Move the negative in front of the fraction.

$y - 9 = \frac{x}{18} - \frac{9}{2}$

Move all terms not containing $y$ to the right side of the equation.

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Add $9$ to both sides of the equation.

$y = \frac{x}{18} - \frac{9}{2} + 9$

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{x}{18} - \frac{9}{2} + 9 \cdot \frac{2}{2}$

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

$y = \frac{x}{18} - \frac{9}{2} + \frac{9 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{x}{18} + \frac{- 9 + 9 \cdot 2}{2}$

Simplify the numerator.

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Multiply $9$ by $2$ .

$y = \frac{x}{18} + \frac{- 9 + 18}{2}$

Add $- 9$ and $18$ .

$y = \frac{x}{18} + \frac{9}{2}$

Rewrite in slope-intercept form.

$y = \frac{1}{18} x + \frac{9}{2}$

Find the Tangent Line at the Point y = square root of x ; (81,9)

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