Calculus Examples

y = sec (x)

$y = \sec (x)$ ,

(\frac{π}{6}, 2 \frac{\sqrt{3}}{3})

$(\frac{π}{6}, 2 \frac{\sqrt{3}}{3})$

Step 1

Find the first derivative and evaluate at

x = \frac{π}{6}

$x = \frac{π}{6}$ and

y = 2 (\frac{\sqrt{3}}{3})

$y = 2 (\frac{\sqrt{3}}{3})$ to find the slope of the tangent line.

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Step 1.1

The derivative of

sec (x)

$\sec (x)$ with respect to

x

$x$ is

sec (x) tan (x)

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

sec (x) tan (x)

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

Step 1.2

Evaluate the derivative at

x = \frac{π}{6}

$x = \frac{π}{6}$ .

sec (\frac{π}{6}) tan (\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})

$\sec (\frac{π}{6})$ is

\frac{2}{\sqrt{3}}

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

\frac{2}{\sqrt{3}} tan (\frac{π}{6})

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

Step 1.3.2

Multiply

\frac{2}{\sqrt{3}}

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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

Step 1.3.3

Combine and simplify the denominator.

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Step 1.3.3.1

Multiply

\frac{2}{\sqrt{3}}

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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

Step 1.3.3.2

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

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

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

Step 1.3.3.3

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

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

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

Step 1.3.3.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 1.3.3.5

Add

1

$1$ and

1

$1$ .

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

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

Step 1.3.3.6

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 1.3.3.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{3}

$\sqrt{3}$ as

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

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

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

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

Step 1.3.3.6.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 1.3.3.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 1.3.3.6.4

Cancel the common factor of

2

$2$ .

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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