Calculus Examples

$f (x) = \tan (2 x)$ , $x = \frac{π}{6}$

Step 1

Find the corresponding $y$ -value to $x = \frac{π}{6}$ .

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Step 1.1

Substitute $\frac{π}{6}$ in for $x$ .

$y = \tan (2 (\frac{π}{6}))$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Multiply $2$ by $\frac{π}{6}$ .

$y = \tan (2 (\frac{π}{6}))$

Step 1.2.2

Simplify $\tan (2 (\frac{π}{6}))$ .

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Factor $2$ out of $6$ .

$y = \tan (2 \frac{π}{2 (3)})$

Step 1.2.2.1.2

Cancel the common factor.

$y = \tan (2 \frac{π}{2 \cdot 3})$

Step 1.2.2.1.3

Rewrite the expression.

$y = \tan (\frac{π}{3})$

Step 1.2.2.2

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

$y = \sqrt{3}$

Step 2

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

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 2 x$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 2.1.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$\sec^{2} (u) \frac{d}{d x} [2 x]$

Step 2.1.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\sec^{2} (2 x) (2 \cdot 1)$

Step 2.2.3

Simplify the expression.

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Step 2.2.3.1

Multiply $2$ by $1$ .

$\sec^{2} (2 x) \cdot 2$

Step 2.2.3.2

Move $2$ to the left of $\sec^{2} (2 x)$ .

$2 \sec^{2} (2 x)$

Step 2.3

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

$2 \sec^{2} (2 (\frac{π}{6}))$

Step 2.4

Simplify.

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Step 2.4.1

Cancel the common factor of $2$ .

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Step 2.4.1.1

Factor $2$ out of $6$ .

$2 \sec^{2} (2 \frac{π}{2 (3)})$

Step 2.4.1.2

Cancel the common factor.

$2 \sec^{2} (2 \frac{π}{2 \cdot 3})$

Step 2.4.1.3

Rewrite the expression.

$2 \sec^{2} (\frac{π}{3})$

Step 2.4.2

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

$2 \cdot 2^{2}$

Step 2.4.3

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 2.4.3.1

Multiply $2$ by $2^{2}$ .

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Step 2.4.3.1.1

Raise $2$ to the power of $1$ .

$2^{1} \cdot 2^{2}$

Step 2.4.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{1 + 2}$

Step 2.4.3.2

Add $1$ and $2$ .

$2^{3}$

Step 2.4.4

Raise $2$ to the power of $3$ .

$8$

Step 3

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

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Step 3.1

Use the slope $8$ and a given point $(\frac{π}{6}, \sqrt{3})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\sqrt{3}) = 8 \cdot (x - (\frac{π}{6}))$

Step 3.2

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

$y - \sqrt{3} = 8 \cdot (x - \frac{π}{6})$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $8 \cdot (x - \frac{π}{6})$ .

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Step 3.3.1.1

Rewrite.

$y - \sqrt{3} = 0 + 0 + 8 \cdot (x - \frac{π}{6})$

Step 3.3.1.2

Simplify by adding zeros.

$y - \sqrt{3} = 8 \cdot (x - \frac{π}{6})$

Step 3.3.1.3

Apply the distributive property.

$y - \sqrt{3} = 8 x + 8 (- \frac{π}{6})$

Step 3.3.1.4

Cancel the common factor of $2$ .

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Step 3.3.1.4.1

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

$y - \sqrt{3} = 8 x + 8 \frac{- π}{6}$

Step 3.3.1.4.2

Factor $2$ out of $8$ .

$y - \sqrt{3} = 8 x + 2 (4) \frac{- π}{6}$

Step 3.3.1.4.3

Factor $2$ out of $6$ .

$y - \sqrt{3} = 8 x + 2 \cdot 4 \frac{- π}{2 \cdot 3}$

Step 3.3.1.4.4

Cancel the common factor.

$y - \sqrt{3} = 8 x + 2 \cdot 4 \frac{- π}{2 \cdot 3}$

Step 3.3.1.4.5

Rewrite the expression.

$y - \sqrt{3} = 8 x + 4 \frac{- π}{3}$

Step 3.3.1.5

Combine $4$ and $\frac{- π}{3}$ .

$y - \sqrt{3} = 8 x + \frac{4 (- π)}{3}$

Step 3.3.1.6

Simplify the expression.

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Step 3.3.1.6.1

Multiply $- 1$ by $4$ .

$y - \sqrt{3} = 8 x + \frac{- 4 π}{3}$

Step 3.3.1.6.2

Move the negative in front of the fraction.

$y - \sqrt{3} = 8 x - \frac{4 π}{3}$

Step 3.3.2

Add $\sqrt{3}$ to both sides of the equation.

$y = 8 x - \frac{4 π}{3} + \sqrt{3}$

Step 3.3.3

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

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Step 3.3.3.1

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

$y = 8 x - \frac{4 π}{3} + \sqrt{3} \cdot \frac{3}{3}$

Step 3.3.3.2

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

$y = 8 x - \frac{4 π}{3} + \frac{\sqrt{3} \cdot 3}{3}$

Step 3.3.3.3

Combine the numerators over the common denominator.

$y = 8 x + \frac{- 4 π + \sqrt{3} \cdot 3}{3}$

Step 3.3.3.4

Move $3$ to the left of $\sqrt{3}$ .

$y = 8 x + \frac{- 4 π + 3 \sqrt{3}}{3}$

Step 3.3.3.5

Factor $- 1$ out of $- 4 π$ .

$y = 8 x + \frac{- (4 π) + 3 \sqrt{3}}{3}$

Step 3.3.3.6

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

$y = 8 x + \frac{- (4 π) - (- 3 \sqrt{3})}{3}$

Step 3.3.3.7

Factor $- 1$ out of $- (4 π) - (- 3 \sqrt{3})$ .

$y = 8 x + \frac{- (4 π - 3 \sqrt{3})}{3}$

Step 3.3.3.8

Rewrite $- (4 π - 3 \sqrt{3})$ as $- 1 (4 π - 3 \sqrt{3})$ .

$y = 8 x + \frac{- 1 (4 π - 3 \sqrt{3})}{3}$

Step 3.3.3.9

Move the negative in front of the fraction.

$y = 8 x - \frac{4 π - 3 \sqrt{3}}{3}$

Step 4