Calculus Examples

$y = \sec (x) - 2 \cos (x)$ , $p = (\frac{π}{3}, 1)$

Step 1

Find the first derivative and evaluate at $x = \frac{π}{3}$ and $y = 1$ to find the slope of the tangent line.

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Step 1.1

By the Sum Rule, the derivative of $\sec (x) - 2 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [- 2 \cos (x)]$ .

$\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [- 2 \cos (x)]$

Step 1.2

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

$\sec (x) \tan (x) + \frac{d}{d x} [- 2 \cos (x)]$

Step 1.3

Evaluate $\frac{d}{d x} [- 2 \cos (x)]$ .

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Step 1.3.1

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

$\sec (x) \tan (x) - 2 \frac{d}{d x} [\cos (x)]$

Step 1.3.2

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

$\sec (x) \tan (x) - 2 (- \sin (x))$

Step 1.3.3

Multiply $- 1$ by $- 2$ .

$\sec (x) \tan (x) + 2 \sin (x)$

Step 1.4

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

$\sec (\frac{π}{3}) \tan (\frac{π}{3}) + 2 \sin (\frac{π}{3})$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

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

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

Step 1.5.1.2

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

$2 \sqrt{3} + 2 \sin (\frac{π}{3})$

Step 1.5.1.3

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

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

Step 1.5.1.4

Cancel the common factor of $2$ .

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Step 1.5.1.4.1

Cancel the common factor.

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

Step 1.5.1.4.2

Rewrite the expression.

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

Step 1.5.2

Add $2 \sqrt{3}$ and $\sqrt{3}$ .

$3 \sqrt{3}$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

Cancel the common factor of $3$ .

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Step 2.3.1.4.1

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

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

Step 2.3.1.4.2

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

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

Step 2.3.1.4.3

Cancel the common factor.

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

Step 2.3.1.4.4

Rewrite the expression.

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

Step 2.3.2

Add $1$ to both sides of the equation.

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

Step 2.3.3

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

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Step 2.3.3.1

Remove parentheses.

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

Step 2.3.3.2

Simplify each term.

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Step 2.3.3.2.1

Move $- 1$ to the left of $\sqrt{3}$ .

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

Step 2.3.3.2.2

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$y = 3 \sqrt{3} x - \sqrt{3} π + 1$

Step 3