Calculus Examples

$y = 6 \sec (x) - 12 \cos (x)$ , $(\frac{π}{3}, 6)$

Step 1

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

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $6 \sec (x) - 12 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [6 \sec (x)] + \frac{d}{d x} [- 12 \cos (x)]$ .

$\frac{d}{d x} [6 \sec (x)] + \frac{d}{d x} [- 12 \cos (x)]$

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

$6 \frac{d}{d x} [\sec (x)] + \frac{d}{d x} [- 12 \cos (x)]$

Step 1.2.2

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

$6 \sec (x) \tan (x) + \frac{d}{d x} [- 12 \cos (x)]$

Step 1.3

Evaluate $\frac{d}{d x} [- 12 \cos (x)]$ .

Tap for more steps...

Step 1.3.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 \cos (x)$ with respect to $x$ is $- 12 \frac{d}{d x} [\cos (x)]$ .

$6 \sec (x) \tan (x) - 12 \frac{d}{d x} [\cos (x)]$

Step 1.3.2

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

$6 \sec (x) \tan (x) - 12 (- \sin (x))$

Step 1.3.3

Multiply $- 1$ by $- 12$ .

$6 \sec (x) \tan (x) + 12 \sin (x)$

Step 1.4

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

$6 \sec (\frac{π}{3}) \tan (\frac{π}{3}) + 12 \sin (\frac{π}{3})$

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 $2$ .

$6 \cdot 2 \tan (\frac{π}{3}) + 12 \sin (\frac{π}{3})$

Step 1.5.1.2

Multiply $6$ by $2$ .

$12 \tan (\frac{π}{3}) + 12 \sin (\frac{π}{3})$

Step 1.5.1.3

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

$12 \sqrt{3} + 12 \sin (\frac{π}{3})$

Step 1.5.1.4

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

$12 \sqrt{3} + 12 \frac{\sqrt{3}}{2}$

Step 1.5.1.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.5.1.5.1

Factor $2$ out of $12$ .

$12 \sqrt{3} + 2 (6) \frac{\sqrt{3}}{2}$

Step 1.5.1.5.2

Cancel the common factor.

$12 \sqrt{3} + 2 \cdot 6 \frac{\sqrt{3}}{2}$

Step 1.5.1.5.3

Rewrite the expression.

$12 \sqrt{3} + 6 \sqrt{3}$

Step 1.5.2

Add $12 \sqrt{3}$ and $6 \sqrt{3}$ .

$18 \sqrt{3}$

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 $18 \sqrt{3}$ and a given point $(\frac{π}{3}, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = 18 \sqrt{3} \cdot (x - (\frac{π}{3}))$

Step 2.2

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

$y - 6 = 18 \sqrt{3} \cdot (x - \frac{π}{3})$

Step 2.3

Solve for $y$ .

Tap for more steps...

Step 2.3.1

Simplify $18 \sqrt{3} \cdot (x - \frac{π}{3})$ .

Tap for more steps...

Step 2.3.1.1

Rewrite.

$y - 6 = 0 + 0 + 18 \sqrt{3} \cdot (x - \frac{π}{3})$

Step 2.3.1.2

Simplify by adding zeros.

$y - 6 = 18 \sqrt{3} \cdot (x - \frac{π}{3})$

Step 2.3.1.3

Apply the distributive property.

$y - 6 = 18 \sqrt{3} x + 18 \sqrt{3} (- \frac{π}{3})$

Step 2.3.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.3.1.4.1

Move the leading negative in $- \frac{π}{3}$ into the numerator.

$y - 6 = 18 \sqrt{3} x + 18 \sqrt{3} \frac{- π}{3}$

Step 2.3.1.4.2

Factor $3$ out of $18 \sqrt{3}$ .

$y - 6 = 18 \sqrt{3} x + 3 (6 \sqrt{3}) \frac{- π}{3}$

Step 2.3.1.4.3

Cancel the common factor.

$y - 6 = 18 \sqrt{3} x + 3 (6 \sqrt{3}) \frac{- π}{3}$

Step 2.3.1.4.4

Rewrite the expression.

$y - 6 = 18 \sqrt{3} x + 6 \sqrt{3} (- π)$

Step 2.3.1.5

Multiply $- 1$ by $6$ .

$y - 6 = 18 \sqrt{3} x - 6 \sqrt{3} π$

Step 2.3.2

Add $6$ to both sides of the equation.

$y = 18 \sqrt{3} x - 6 \sqrt{3} π + 6$

Step 3