Calculus Examples

$y = \sec (x)$ , $(\frac{π}{3}, 2)$

Step 1

Find the first derivative and evaluate at $x = \frac{π}{3}$ and $y = 2$ 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 $x = \frac{π}{3}$ .

$\sec (\frac{π}{3}) \tan (\frac{π}{3})$

Step 1.3

Simplify.

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Step 1.3.1

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

$2 \tan (\frac{π}{3})$

Step 1.3.2

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

$2 \sqrt{3}$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

Multiply $2 \sqrt{3} (- \frac{π}{3})$ .

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Step 2.3.1.4.1

Multiply $- 1$ by $2$ .

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

Step 2.3.1.4.2

Combine $\frac{π}{3}$ and $- 2$ .

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

Step 2.3.1.4.3

Combine $\frac{π \cdot - 2}{3}$ and $\sqrt{3}$ .

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

Step 2.3.1.5

Simplify each term.

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Step 2.3.1.5.1

Move $- 2$ to the left of $π$ .

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

Step 2.3.1.5.2

Move the negative in front of the fraction.

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

Step 2.3.2

Add $2$ to both sides of the equation.

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

Step 2.3.3

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

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Step 2.3.3.1

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

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

Step 2.3.3.2

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

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

Step 2.3.3.3

Combine the numerators over the common denominator.

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

Step 2.3.3.4

Multiply $2$ by $3$ .

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

Step 2.3.3.5

Factor $- 1$ out of $- 2 π \sqrt{3}$ .

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

Step 2.3.3.6

Rewrite $6$ as $- 1 (- 6)$ .

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

Step 2.3.3.7

Factor $- 1$ out of $- (2 π \sqrt{3}) - 1 (- 6)$ .

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

Step 2.3.3.8

Rewrite $- (2 π \sqrt{3} - 6)$ as $- 1 (2 π \sqrt{3} - 6)$ .

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

Step 2.3.3.9

Move the negative in front of the fraction.

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

Step 3