Calculus Examples

$y = \sec (x) - 6 \cos (x)$ , $P = (\frac{π}{3}, - 1)$

Step 1

Find the first derivative and evaluate at $x = \frac{π}{3}$ and $y = - 1$ to find the slope of the tangent line.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $\sec (x) - 6 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [- 6 \cos (x)]$ .

$\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [- 6 \cos (x)]$

Step 1.2

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

$\sec (x) \tan (x) + \frac{d}{d x} [- 6 \cos (x)]$

Step 1.3

Evaluate $\frac{d}{d x} [- 6 \cos (x)]$ .

Tap for more steps...

Step 1.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \cos (x)$ with respect to $x$ is $- 6 \frac{d}{d x} [\cos (x)]$ .

$\sec (x) \tan (x) - 6 \frac{d}{d x} [\cos (x)]$

Step 1.3.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$\sec (x) \tan (x) - 6 (- \sin (x))$

Step 1.3.3

Multiply $- 1$ by $- 6$ .

$\sec (x) \tan (x) + 6 \sin (x)$

Step 1.4

Evaluate the derivative at $x = \frac{π}{3}$ .

$\sec (\frac{π}{3}) \tan (\frac{π}{3}) + 6 \sin (\frac{π}{3})$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Simplify each term.

Tap for more steps...

Step 1.5.1.1

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

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

Step 1.5.1.2

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

$2 \sqrt{3} + 6 \sin (\frac{π}{3})$

Step 1.5.1.3

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

$2 \sqrt{3} + 6 \frac{\sqrt{3}}{2}$

Step 1.5.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.5.1.4.1

Factor $2$ out of $6$ .

$2 \sqrt{3} + 2 (3) \frac{\sqrt{3}}{2}$

Step 1.5.1.4.2

Cancel the common factor.

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

Step 1.5.1.4.3

Rewrite the expression.

$2 \sqrt{3} + 3 \sqrt{3}$

Step 1.5.2

Add $2 \sqrt{3}$ and $3 \sqrt{3}$ .

$5 \sqrt{3}$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

Tap for more steps...

Step 2.1

Use the slope $5 \sqrt{3}$ and a given point $P = (\frac{π}{3}, - 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 1) = 5 \sqrt{3} \cdot (x - (\frac{π}{3}))$

Step 2.2

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

$y + 1 = 5 \sqrt{3} \cdot (x - \frac{π}{3})$

Step 2.3

Solve for $y$ .

Tap for more steps...

Step 2.3.1

Simplify $5 \sqrt{3} \cdot (x - \frac{π}{3})$ .

Tap for more steps...

Step 2.3.1.1

Rewrite.

$y + 1 = 0 + 0 + 5 \sqrt{3} \cdot (x - \frac{π}{3})$

Step 2.3.1.2

Simplify by adding zeros.

$y + 1 = 5 \sqrt{3} \cdot (x - \frac{π}{3})$

Step 2.3.1.3

Apply the distributive property.

$y + 1 = 5 \sqrt{3} x + 5 \sqrt{3} (- \frac{π}{3})$

Step 2.3.1.4

Multiply $5 \sqrt{3} (- \frac{π}{3})$ .

Tap for more steps...

Step 2.3.1.4.1

Multiply $- 1$ by $5$ .

$y + 1 = 5 \sqrt{3} x - 5 \sqrt{3} \frac{π}{3}$

Step 2.3.1.4.2

Combine $\frac{π}{3}$ and $- 5$ .

$y + 1 = 5 \sqrt{3} x + \frac{π \cdot - 5}{3} \sqrt{3}$

Step 2.3.1.4.3

Combine $\frac{π \cdot - 5}{3}$ and $\sqrt{3}$ .

$y + 1 = 5 \sqrt{3} x + \frac{π \cdot - 5 \sqrt{3}}{3}$

Step 2.3.1.5

Simplify each term.

Tap for more steps...

Step 2.3.1.5.1

Move $- 5$ to the left of $π$ .

$y + 1 = 5 \sqrt{3} x + \frac{- 5 \cdot π \sqrt{3}}{3}$

Step 2.3.1.5.2

Move the negative in front of the fraction.

$y + 1 = 5 \sqrt{3} x - \frac{5 π \sqrt{3}}{3}$

Step 2.3.2

Subtract $1$ from both sides of the equation.

$y = 5 \sqrt{3} x - \frac{5 π \sqrt{3}}{3} - 1$

Step 2.3.3

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

Tap for more steps...

Step 2.3.3.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = 5 \sqrt{3} x - \frac{5 π \sqrt{3}}{3} - 1 \cdot \frac{3}{3}$

Step 2.3.3.2

Combine $- 1$ and $\frac{3}{3}$ .

$y = 5 \sqrt{3} x - \frac{5 π \sqrt{3}}{3} + \frac{- 1 \cdot 3}{3}$

Step 2.3.3.3

Combine the numerators over the common denominator.

$y = 5 \sqrt{3} x + \frac{- 5 π \sqrt{3} - 1 \cdot 3}{3}$

Step 2.3.3.4

Multiply $- 1$ by $3$ .

$y = 5 \sqrt{3} x + \frac{- 5 π \sqrt{3} - 3}{3}$

Step 2.3.3.5

Factor $- 1$ out of $- 5 π \sqrt{3}$ .

$y = 5 \sqrt{3} x + \frac{- (5 π \sqrt{3}) - 3}{3}$

Step 2.3.3.6

Rewrite $- 3$ as $- 1 (3)$ .

$y = 5 \sqrt{3} x + \frac{- (5 π \sqrt{3}) - 1 (3)}{3}$

Step 2.3.3.7

Factor $- 1$ out of $- (5 π \sqrt{3}) - 1 (3)$ .

$y = 5 \sqrt{3} x + \frac{- (5 π \sqrt{3} + 3)}{3}$

Step 2.3.3.8

Rewrite $- (5 π \sqrt{3} + 3)$ as $- 1 (5 π \sqrt{3} + 3)$ .

$y = 5 \sqrt{3} x + \frac{- 1 (5 π \sqrt{3} + 3)}{3}$

Step 2.3.3.9

Move the negative in front of the fraction.

$y = 5 \sqrt{3} x - \frac{5 π \sqrt{3} + 3}{3}$

Step 3