Calculus Examples

$y = 8 \sin (x)$ at the point $(\frac{π}{6}, 4)$

Step 1

Find the first derivative and evaluate at $x = \frac{π}{6}$ and $y = 4$ to find the slope of the tangent line.

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Step 1.1

Since $8$ is constant with respect to $x$ , the derivative of $8 \sin (x)$ with respect to $x$ is $8 \frac{d}{d x} [\sin (x)]$ .

$8 \frac{d}{d x} [\sin (x)]$

Step 1.2

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

$8 \cos (x)$

Step 1.3

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

$8 \cos (\frac{π}{6})$

Step 1.4

Simplify.

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Step 1.4.1

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

$8 \frac{\sqrt{3}}{2}$

Step 1.4.2

Cancel the common factor of $2$ .

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Step 1.4.2.1

Factor $2$ out of $8$ .

$2 (4) \frac{\sqrt{3}}{2}$

Step 1.4.2.2

Cancel the common factor.

$2 \cdot 4 \frac{\sqrt{3}}{2}$

Step 1.4.2.3

Rewrite the expression.

$4 \sqrt{3}$

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 $4 \sqrt{3}$ and a given point $(\frac{π}{6}, 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4) = 4 \sqrt{3} \cdot (x - (\frac{π}{6}))$

Step 2.2

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

$y - 4 = 4 \sqrt{3} \cdot (x - \frac{π}{6})$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $4 \sqrt{3} \cdot (x - \frac{π}{6})$ .

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Step 2.3.1.1

Rewrite.

$y - 4 = 0 + 0 + 4 \sqrt{3} \cdot (x - \frac{π}{6})$

Step 2.3.1.2

Simplify by adding zeros.

$y - 4 = 4 \sqrt{3} \cdot (x - \frac{π}{6})$

Step 2.3.1.3

Apply the distributive property.

$y - 4 = 4 \sqrt{3} x + 4 \sqrt{3} (- \frac{π}{6})$

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

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

$y - 4 = 4 \sqrt{3} x + 4 \sqrt{3} \frac{- π}{6}$

Step 2.3.1.4.2

Factor $2$ out of $4 \sqrt{3}$ .

$y - 4 = 4 \sqrt{3} x + 2 (2 \sqrt{3}) \frac{- π}{6}$

Step 2.3.1.4.3

Factor $2$ out of $6$ .

$y - 4 = 4 \sqrt{3} x + 2 (2 \sqrt{3}) \frac{- π}{2 (3)}$

Step 2.3.1.4.4

Cancel the common factor.

$y - 4 = 4 \sqrt{3} x + 2 (2 \sqrt{3}) \frac{- π}{2 \cdot 3}$

Step 2.3.1.4.5

Rewrite the expression.

$y - 4 = 4 \sqrt{3} x + 2 \sqrt{3} \frac{- π}{3}$

Step 2.3.1.5

Combine $\frac{- π}{3}$ and $2$ .

$y - 4 = 4 \sqrt{3} x + \frac{- π \cdot 2}{3} \sqrt{3}$

Step 2.3.1.6

Multiply $2$ by $- 1$ .

$y - 4 = 4 \sqrt{3} x + \frac{- 2 π}{3} \sqrt{3}$

Step 2.3.1.7

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

$y - 4 = 4 \sqrt{3} x + \frac{- 2 π \sqrt{3}}{3}$

Step 2.3.1.8

Move the negative in front of the fraction.

$y - 4 = 4 \sqrt{3} x - \frac{2 π \sqrt{3}}{3}$

Step 2.3.2

Add $4$ to both sides of the equation.

$y = 4 \sqrt{3} x - \frac{2 π \sqrt{3}}{3} + 4$

Step 2.3.3

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

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Step 2.3.3.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = 4 \sqrt{3} x - \frac{2 π \sqrt{3}}{3} + 4 \cdot \frac{3}{3}$

Step 2.3.3.2

Combine $4$ and $\frac{3}{3}$ .

$y = 4 \sqrt{3} x - \frac{2 π \sqrt{3}}{3} + \frac{4 \cdot 3}{3}$

Step 2.3.3.3

Combine the numerators over the common denominator.

$y = 4 \sqrt{3} x + \frac{- 2 π \sqrt{3} + 4 \cdot 3}{3}$

Step 2.3.3.4

Multiply $4$ by $3$ .

$y = 4 \sqrt{3} x + \frac{- 2 π \sqrt{3} + 12}{3}$

Step 2.3.3.5

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

$y = 4 \sqrt{3} x + \frac{- (2 π \sqrt{3}) + 12}{3}$

Step 2.3.3.6

Rewrite $12$ as $- 1 (- 12)$ .

$y = 4 \sqrt{3} x + \frac{- (2 π \sqrt{3}) - 1 (- 12)}{3}$

Step 2.3.3.7

Factor $- 1$ out of $- (2 π \sqrt{3}) - 1 (- 12)$ .

$y = 4 \sqrt{3} x + \frac{- (2 π \sqrt{3} - 12)}{3}$

Step 2.3.3.8

Rewrite $- (2 π \sqrt{3} - 12)$ as $- 1 (2 π \sqrt{3} - 12)$ .

$y = 4 \sqrt{3} x + \frac{- 1 (2 π \sqrt{3} - 12)}{3}$

Step 2.3.3.9

Move the negative in front of the fraction.

$y = 4 \sqrt{3} x - \frac{2 π \sqrt{3} - 12}{3}$

Step 3