Calculus Examples

$y = \sec (x)$ , $(\frac{π}{6}, 2 \frac{\sqrt{3}}{3})$

Step 1

Find the first derivative and evaluate at $x = \frac{π}{6}$ and $y = 2 (\frac{\sqrt{3}}{3})$ to find the slope of the tangent line.

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Step 1.1

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\sec (x) \tan (x)$

Step 1.2

Evaluate the derivative at $x = \frac{π}{6}$ .

$\sec (\frac{π}{6}) \tan (\frac{π}{6})$

Step 1.3

Simplify.

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Step 1.3.1

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

$\frac{2}{\sqrt{3}} \tan (\frac{π}{6})$

Step 1.3.2

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \tan (\frac{π}{6})$

Step 1.3.3

Combine and simplify the denominator.

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Step 1.3.3.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} \tan (\frac{π}{6})$

Step 1.3.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \tan (\frac{π}{6})$

Step 1.3.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \tan (\frac{π}{6})$

Step 1.3.3.4

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

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} \tan (\frac{π}{6})$

Step 1.3.3.5

Add $1$ and $1$ .

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} \tan (\frac{π}{6})$

Step 1.3.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.3.3.6.1

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

$\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \tan (\frac{π}{6})$

Step 1.3.3.6.2

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

$\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \tan (\frac{π}{6})$

Step 1.3.3.6.3

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

$\frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \tan (\frac{π}{6})$

Step 1.3.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.3.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \tan (\frac{π}{6})$

Step 1.3.3.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{3}}{3^{1}} \tan (\frac{π}{6})$

Step 1.3.3.6.5

Evaluate the exponent.

$\frac{2 \sqrt{3}}{3} \tan (\frac{π}{6})$

Step 1.3.4

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

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

Step 1.3.5

Multiply $\frac{2 \sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3}$ .

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Step 1.3.5.1

Multiply $\frac{2 \sqrt{3}}{3}$ by $\frac{\sqrt{3}}{3}$ .

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

Step 1.3.5.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 1.3.5.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 1.3.5.4

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

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

Step 1.3.5.5

Add $1$ and $1$ .

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

Step 1.3.5.6

Multiply $3$ by $3$ .

$\frac{2 {\sqrt{3}}^{2}}{9}$

Step 1.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.3.6.1

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

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

Step 1.3.6.2

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

$\frac{2 \cdot 3^{\frac{1}{2} \cdot 2}}{9}$

Step 1.3.6.3

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

$\frac{2 \cdot 3^{\frac{2}{2}}}{9}$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\frac{2 \cdot 3^{\frac{2}{2}}}{9}$

Step 1.3.6.4.2

Rewrite the expression.

$\frac{2 \cdot 3^{1}}{9}$

Step 1.3.6.5

Evaluate the exponent.

$\frac{2 \cdot 3}{9}$

Step 1.3.7

Multiply $2$ by $3$ .

$\frac{6}{9}$

Step 1.3.8

Cancel the common factor of $6$ and $9$ .

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Step 1.3.8.1

Factor $3$ out of $6$ .

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

Step 1.3.8.2

Cancel the common factors.

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Step 1.3.8.2.1

Factor $3$ out of $9$ .

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

Step 1.3.8.2.2

Cancel the common factor.

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

Step 1.3.8.2.3

Rewrite the expression.

$\frac{2}{3}$

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

$y - (2 (\frac{\sqrt{3}}{3})) = \frac{2}{3} \cdot (x - (\frac{π}{6}))$

Step 2.2

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

$y - \frac{2 \sqrt{3}}{3} = \frac{2}{3} \cdot (x - \frac{π}{6})$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{2}{3} \cdot (x - \frac{π}{6})$ .

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Step 2.3.1.1

Rewrite.

$y - \frac{2 \sqrt{3}}{3} = 0 + 0 + \frac{2}{3} \cdot (x - \frac{π}{6})$

Step 2.3.1.2

Simplify by adding zeros.

$y - \frac{2 \sqrt{3}}{3} = \frac{2}{3} \cdot (x - \frac{π}{6})$

Step 2.3.1.3

Apply the distributive property.

$y - \frac{2 \sqrt{3}}{3} = \frac{2}{3} x + \frac{2}{3} (- \frac{π}{6})$

Step 2.3.1.4

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

$y - \frac{2 \sqrt{3}}{3} = \frac{2 x}{3} + \frac{2}{3} (- \frac{π}{6})$

Step 2.3.1.5

Cancel the common factor of $2$ .

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Step 2.3.1.5.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

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

Step 2.3.1.5.2

Factor $2$ out of $6$ .

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

Step 2.3.1.5.3

Cancel the common factor.

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

Step 2.3.1.5.4

Rewrite the expression.

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

Step 2.3.1.6

Multiply $\frac{1}{3}$ by $\frac{- π}{3}$ .

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

Step 2.3.1.7

Simplify the expression.

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Step 2.3.1.7.1

Multiply $3$ by $3$ .

$y - \frac{2 \sqrt{3}}{3} = \frac{2 x}{3} + \frac{- π}{9}$

Step 2.3.1.7.2

Move the negative in front of the fraction.

$y - \frac{2 \sqrt{3}}{3} = \frac{2 x}{3} - \frac{π}{9}$

Step 2.3.2

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

$y = \frac{2 x}{3} - \frac{π}{9} + \frac{2 \sqrt{3}}{3}$

Step 2.3.3

Write in $y = m x + b$ form.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.3.2.1

Multiply $\frac{2 \sqrt{3}}{3}$ by $\frac{3}{3}$ .

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

Step 2.3.3.2.2

Multiply $3$ by $3$ .

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

Step 2.3.3.3

Combine the numerators over the common denominator.

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

Step 2.3.3.4

Multiply $3$ by $2$ .

$y = \frac{2 x}{3} + \frac{- π + 6 \sqrt{3}}{9}$

Step 2.3.3.5

Factor $- 1$ out of $- π$ .

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

Step 2.3.3.6

Factor $- 1$ out of $6 \sqrt{3}$ .

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

Step 2.3.3.7

Factor $- 1$ out of $- (π) - (- 6 \sqrt{3})$ .

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

Step 2.3.3.8

Rewrite $- (π - 6 \sqrt{3})$ as $- 1 (π - 6 \sqrt{3})$ .

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

Step 2.3.3.9

Move the negative in front of the fraction.

$y = \frac{2 x}{3} - \frac{π - 6 \sqrt{3}}{9}$

Step 2.3.3.10

Reorder terms.

$y = \frac{2}{3} x - \frac{π - 6 \sqrt{3}}{9}$

Step 3