Calculus Examples

$f (x) = - 3 \cos (x) - 2 + \frac{3 \sqrt{2} π}{8} + \frac{3 \sqrt{2}}{2}$ at $x = \frac{π}{4}$

Step 1

Find the corresponding $y$ -value to $x = \frac{π}{4}$ .

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Step 1.1

Substitute $\frac{π}{4}$ in for $x$ .

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

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

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

Step 1.2.2

Simplify $- 3 \cos (\frac{π}{4}) - 2 + \frac{3 \sqrt{2} π}{8} + \frac{3 \sqrt{2}}{2}$ .

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 1.2.2.1.2

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

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

Step 1.2.2.1.3

Move the negative in front of the fraction.

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

Step 1.2.2.2

Simplify terms.

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Step 1.2.2.2.1

Combine the numerators over the common denominator.

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

Step 1.2.2.2.2

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

$y = - 2 + \frac{0}{2} + \frac{3 \sqrt{2} π}{8}$

Step 1.2.2.2.3

Simplify the expression.

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Step 1.2.2.2.3.1

Divide $0$ by $2$ .

$y = - 2 + 0 + \frac{3 \sqrt{2} π}{8}$

Step 1.2.2.2.3.2

Add $- 2$ and $0$ .

$y = - 2 + \frac{3 \sqrt{2} π}{8}$

Step 2

Find the first derivative and evaluate at $x = \frac{π}{4}$ and $y = - 2 + \frac{3 \sqrt{2} π}{8}$ to find the slope of the tangent line.

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Step 2.1

By the Sum Rule, the derivative of $- 3 \cos (x) - 2 + \frac{3 \sqrt{2} π}{8} + \frac{3 \sqrt{2}}{2}$ with respect to $x$ is $\frac{d}{d x} [- 3 \cos (x)] + \frac{d}{d x} [- 2] + \frac{d}{d x} [\frac{3 \sqrt{2} π}{8}] + \frac{d}{d x} [\frac{3 \sqrt{2}}{2}]$ .

$\frac{d}{d x} [- 3 \cos (x)] + \frac{d}{d x} [- 2] + \frac{d}{d x} [\frac{3 \sqrt{2} π}{8}] + \frac{d}{d x} [\frac{3 \sqrt{2}}{2}]$

Step 2.2

Evaluate $\frac{d}{d x} [- 3 \cos (x)]$ .

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Step 2.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 \cos (x)$ with respect to $x$ is $- 3 \frac{d}{d x} [\cos (x)]$ .

$- 3 \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- 2] + \frac{d}{d x} [\frac{3 \sqrt{2} π}{8}] + \frac{d}{d x} [\frac{3 \sqrt{2}}{2}]$

Step 2.2.2

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

$- 3 (- \sin (x)) + \frac{d}{d x} [- 2] + \frac{d}{d x} [\frac{3 \sqrt{2} π}{8}] + \frac{d}{d x} [\frac{3 \sqrt{2}}{2}]$

Step 2.2.3

Multiply $- 1$ by $- 3$ .

$3 \sin (x) + \frac{d}{d x} [- 2] + \frac{d}{d x} [\frac{3 \sqrt{2} π}{8}] + \frac{d}{d x} [\frac{3 \sqrt{2}}{2}]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$3 \sin (x) + 0 + \frac{d}{d x} [\frac{3 \sqrt{2} π}{8}] + \frac{d}{d x} [\frac{3 \sqrt{2}}{2}]$

Step 2.3.2

Since $\frac{3 \sqrt{2} π}{8}$ is constant with respect to $x$ , the derivative of $\frac{3 \sqrt{2} π}{8}$ with respect to $x$ is $0$ .

$3 \sin (x) + 0 + 0 + \frac{d}{d x} [\frac{3 \sqrt{2}}{2}]$

Step 2.3.3

Since $\frac{3 \sqrt{2}}{2}$ is constant with respect to $x$ , the derivative of $\frac{3 \sqrt{2}}{2}$ with respect to $x$ is $0$ .

$3 \sin (x) + 0 + 0 + 0$

Step 2.4

Combine terms.

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Step 2.4.1

Add $3 \sin (x)$ and $0$ .

$3 \sin (x) + 0 + 0$

Step 2.4.2

Add $3 \sin (x)$ and $0$ .

$3 \sin (x) + 0$

Step 2.4.3

Add $3 \sin (x)$ and $0$ .

$3 \sin (x)$

Step 2.5

Evaluate the derivative at $x = \frac{π}{4}$ .

$3 \sin (\frac{π}{4})$

Step 2.6

Simplify.

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Step 2.6.1

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 2.6.2

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

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

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 $\frac{3 \sqrt{2}}{2}$ and a given point $(\frac{π}{4}, - 2 + \frac{3 \sqrt{2} π}{8})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 3.2

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

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

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $\frac{3 \sqrt{2}}{2} \cdot (x - \frac{π}{4})$ .

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Step 3.3.1.1

Rewrite.

$y + 2 - \frac{3 \sqrt{2} π}{8} = 0 + 0 + \frac{3 \sqrt{2}}{2} \cdot (x - \frac{π}{4})$

Step 3.3.1.2

Simplify by adding zeros.

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

Step 3.3.1.3

Apply the distributive property.

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

Step 3.3.1.4

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

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

Step 3.3.1.5

Multiply $\frac{3 \sqrt{2}}{2} (- \frac{π}{4})$ .

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Step 3.3.1.5.1

Multiply $\frac{3 \sqrt{2}}{2}$ by $\frac{π}{4}$ .

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

Step 3.3.1.5.2

Multiply $2$ by $4$ .

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

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.2.1

Subtract $2$ from both sides of the equation.

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

Step 3.3.2.2

Add $\frac{3 \sqrt{2} π}{8}$ to both sides of the equation.

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

Step 3.3.2.3

Combine the numerators over the common denominator.

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

Step 3.3.2.4

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

$y = - 2 + \frac{3 \sqrt{2} x}{2} + \frac{0}{8}$

Step 3.3.2.5

Combine the opposite terms in $- 2 + \frac{3 \sqrt{2} x}{2} + \frac{0}{8}$ .

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Step 3.3.2.5.1

Divide $0$ by $8$ .

$y = - 2 + \frac{3 \sqrt{2} x}{2} + 0$

Step 3.3.2.5.2

Add $- 2 + \frac{3 \sqrt{2} x}{2}$ and $0$ .

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

Step 3.3.3

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

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Step 3.3.3.1

Reorder $- 2$ and $\frac{3 \sqrt{2} x}{2}$ .

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

Step 3.3.3.2

Reorder terms.

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

Step 4