Calculus Examples

$y = 2 \sin (x)$ at the point $(\frac{π}{6}, 1)$

Step 1

Find the first derivative and evaluate at $x = \frac{π}{6}$ and $y = 1$ to find the slope of the tangent line.

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Step 1.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \sin (x)$ with respect to $x$ is $2 \frac{d}{d x} [\sin (x)]$ .

$2 \frac{d}{d x} [\sin (x)]$

Step 1.2

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

$2 \cos (x)$

Step 1.3

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

$2 \cos (\frac{π}{6})$

Step 1.4

Simplify.

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Step 1.4.1

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

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

Step 1.4.2

Cancel the common factor of $2$ .

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Step 1.4.2.1

Cancel the common factor.

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

Step 1.4.2.2

Rewrite the expression.

$\sqrt{3}$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 + \sqrt{3} \cdot (x - \frac{π}{6})$

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

Combine $\sqrt{3}$ and $\frac{π}{6}$ .

$y - 1 = \sqrt{3} x - \frac{\sqrt{3} π}{6}$

Step 2.3.2

Add $1$ to both sides of the equation.

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

Step 2.3.3

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

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Step 2.3.3.1

Write $1$ as a fraction with a common denominator.

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

Step 2.3.3.2

Combine the numerators over the common denominator.

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

Step 2.3.3.3

Factor $- 1$ out of $- \sqrt{3} π$ .

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

Step 2.3.3.4

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

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

Step 2.3.3.5

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

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

Step 2.3.3.6

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

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

Step 2.3.3.7

Move the negative in front of the fraction.

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

Step 3