Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 3 x$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $3 x$ .

$\frac{d}{d u} [\cos (u)] \frac{d}{d x} [3 x]$

Step 1.1.2

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

$- \sin (u) \frac{d}{d x} [3 x]$

Step 1.1.3

Replace all occurrences of $u$ with $3 x$ .

$- \sin (3 x) \frac{d}{d x} [3 x]$

Step 1.2

Differentiate.

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Step 1.2.1

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

$- \sin (3 x) (3 \frac{d}{d x} [x])$

Step 1.2.2

Multiply $3$ by $- 1$ .

$- 3 \sin (3 x) \frac{d}{d x} [x]$

Step 1.2.3

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

$- 3 \sin (3 x) \cdot 1$

Step 1.2.4

Multiply $- 3$ by $1$ .

$- 3 \sin (3 x)$

Step 1.3

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

$- 3 \sin (3 (\frac{π}{4}))$

Step 1.4

Simplify.

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Step 1.4.1

Combine $3$ and $\frac{π}{4}$ .

$- 3 \sin (\frac{3 π}{4})$

Step 1.4.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- 3 \sin (\frac{π}{4})$

Step 1.4.3

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

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

Step 1.4.4

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

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

Step 1.4.5

Move the negative in front of the fraction.

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

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

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

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

Step 2.3.1.5

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

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Step 2.3.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.1.5.2

Multiply $\frac{3 \sqrt{2}}{2}$ by $1$ .

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

Step 2.3.1.5.3

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

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

Step 2.3.1.5.4

Multiply $2$ by $4$ .

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

Step 2.3.1.6

Move $3$ to the left of $x$ .

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

Step 2.3.2

Subtract $\frac{\sqrt{2}}{2}$ from both sides of the equation.

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

Step 2.3.3

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

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Step 2.3.3.1

To write $- \frac{\sqrt{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 2.3.3.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

Multiply $2$ by $4$ .

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

Step 2.3.3.3

Combine the numerators over the common denominator.

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

Step 2.3.3.4

Multiply $4$ by $- 1$ .

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

Step 2.3.3.5

Reorder terms.

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

Step 2.3.3.6

Remove parentheses.

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

Step 3