Calculus Examples

$y = 6 \sin (x)$ at the point $(\frac{π}{6}, 3)$

Step 1

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

Tap for more steps...

Step 1.1

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

$6 \frac{d}{d x} [\sin (x)]$

Step 1.2

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

$6 \cos (x)$

Step 1.3

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

$6 \cos (\frac{π}{6})$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

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

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

Step 1.4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.2.1

Factor $2$ out of $6$ .

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

$3 \sqrt{3}$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify terms.

Tap for more steps...

Step 2.3.1.2.1

Apply the distributive property.

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

Step 2.3.1.2.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.3.1.2.2.1

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

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

Step 2.3.1.2.2.2

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

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

Step 2.3.1.2.2.3

Factor $3$ out of $6$ .

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

Step 2.3.1.2.2.4

Cancel the common factor.

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

Step 2.3.1.2.2.5

Rewrite the expression.

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

Step 2.3.1.2.3

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

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

Step 2.3.1.3

Simplify each term.

Tap for more steps...

Step 2.3.1.3.1

Factor out negative.

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

Step 2.3.1.3.2

Move the negative in front of the fraction.

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

Step 2.3.2

Add $3$ to both sides of the equation.

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

Step 2.3.3

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

Tap for more steps...

Step 2.3.3.1

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

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

Step 2.3.3.2

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

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

Step 2.3.3.3

Combine the numerators over the common denominator.

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

Step 2.3.3.4

Multiply $3$ by $2$ .

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

Step 2.3.3.5

Factor $- 1$ out of $- \sqrt{3} π$ .

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

Step 2.3.3.6

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

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

Step 2.3.3.7

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

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

Step 2.3.3.8

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

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

Step 2.3.3.9

Move the negative in front of the fraction.

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

Step 3