Calculus Examples

$f (x) = \cos (\frac{4 x}{3})$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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 $g (x) = \frac{4 x}{3}$ .

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Step 1.1.1.1

To apply the Chain Rule, set $u$ as $\frac{4 x}{3}$ .

$\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{4 x}{3}]$

Step 1.1.1.2

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

$- \sin (u) \frac{d}{d x} [\frac{4 x}{3}]$

Step 1.1.1.3

Replace all occurrences of $u$ with $\frac{4 x}{3}$ .

$- \sin (\frac{4 x}{3}) \frac{d}{d x} [\frac{4 x}{3}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

Since $\frac{4}{3}$ is constant with respect to $x$ , the derivative of $\frac{4 x}{3}$ with respect to $x$ is $\frac{4}{3} \frac{d}{d x} [x]$ .

$- \sin (\frac{4 x}{3}) (\frac{4}{3} \frac{d}{d x} [x])$

Step 1.1.2.2

Combine $\frac{4}{3}$ and $\sin (\frac{4 x}{3})$ .

$- \frac{4 \sin (\frac{4 x}{3})}{3} \frac{d}{d x} [x]$

Step 1.1.2.3

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

$- \frac{4 \sin (\frac{4 x}{3})}{3} \cdot 1$

Step 1.1.2.4

Multiply $- 1$ by $1$ .

$f' (x) = - \frac{4 \sin (\frac{4 x}{3})}{3}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{4 \sin (\frac{4 x}{3})}{3}$ .

$- \frac{4 \sin (\frac{4 x}{3})}{3}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{4 \sin (\frac{4 x}{3})}{3} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{4 \sin (\frac{4 x}{3})}{3} = 0$

Step 2.2

Set the numerator equal to zero.

$4 \sin (\frac{4 x}{3}) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $4 \sin (\frac{4 x}{3}) = 0$ by $4$ and simplify.

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Step 2.3.1.1

Divide each term in $4 \sin (\frac{4 x}{3}) = 0$ by $4$ .

$\frac{4 \sin (\frac{4 x}{3})}{4} = \frac{0}{4}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $4$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{4 \sin (\frac{4 x}{3})}{4} = \frac{0}{4}$

Step 2.3.1.2.1.2

Divide $\sin (\frac{4 x}{3})$ by $1$ .

$\sin (\frac{4 x}{3}) = \frac{0}{4}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $4$ .

$\sin (\frac{4 x}{3}) = 0$

Step 2.3.2

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$\frac{4 x}{3} = \arcsin (0)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

The exact value of $\arcsin (0)$ is $0$ .

$\frac{4 x}{3} = 0$

Step 2.3.4

Set the numerator equal to zero.

$4 x = 0$

Step 2.3.5

Divide each term in $4 x = 0$ by $4$ and simplify.

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Step 2.3.5.1

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Step 2.3.5.2

Simplify the left side.

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Step 2.3.5.2.1

Cancel the common factor of $4$ .

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Step 2.3.5.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Step 2.3.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Step 2.3.5.3

Simplify the right side.

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Step 2.3.5.3.1

Divide $0$ by $4$ .

$x = 0$

Step 2.3.6

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$\frac{4 x}{3} = π - 0$

Step 2.3.7

Solve for $x$ .

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Step 2.3.7.1

Multiply both sides of the equation by $\frac{3}{4}$ .

$\frac{3}{4} \cdot \frac{4 x}{3} = \frac{3}{4} (π - 0)$

Step 2.3.7.2

Simplify both sides of the equation.

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Step 2.3.7.2.1

Simplify the left side.

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Step 2.3.7.2.1.1

Simplify $\frac{3}{4} \cdot \frac{4 x}{3}$ .

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Step 2.3.7.2.1.1.1

Cancel the common factor of $3$ .

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Step 2.3.7.2.1.1.1.1

Cancel the common factor.

$\frac{3}{4} \cdot \frac{4 x}{3} = \frac{3}{4} (π - 0)$

Step 2.3.7.2.1.1.1.2

Rewrite the expression.

$\frac{1}{4} (4 x) = \frac{3}{4} (π - 0)$

Step 2.3.7.2.1.1.2

Cancel the common factor of $4$ .

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Step 2.3.7.2.1.1.2.1

Factor $4$ out of $4 x$ .

$\frac{1}{4} (4 (x)) = \frac{3}{4} (π - 0)$

Step 2.3.7.2.1.1.2.2

Cancel the common factor.

$\frac{1}{4} (4 x) = \frac{3}{4} (π - 0)$

Step 2.3.7.2.1.1.2.3

Rewrite the expression.

$x = \frac{3}{4} (π - 0)$

Step 2.3.7.2.2

Simplify the right side.

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Step 2.3.7.2.2.1

Simplify $\frac{3}{4} (π - 0)$ .

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Step 2.3.7.2.2.1.1

Subtract $0$ from $π$ .

$x = \frac{3}{4} π$

Step 2.3.7.2.2.1.2

Combine $\frac{3}{4}$ and $π$ .

$x = \frac{3 π}{4}$

Step 2.3.8

Find the period of $\sin (\frac{4 x}{3})$ .

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Step 2.3.8.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 2.3.8.2

Replace $b$ with $\frac{4}{3}$ in the formula for period.

$\frac{2 π}{| \frac{4}{3} |}$

Step 2.3.8.3

$\frac{4}{3}$ is approximately $1.\overline{3}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{4}{3}}$

Step 2.3.8.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{4}$

Step 2.3.8.5

Cancel the common factor of $2$ .

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Step 2.3.8.5.1

Factor $2$ out of $2 π$ .

$2 (π) \frac{3}{4}$

Step 2.3.8.5.2

Factor $2$ out of $4$ .

$2 (π) \frac{3}{2 (2)}$

Step 2.3.8.5.3

Cancel the common factor.

$2 π \frac{3}{2 \cdot 2}$

Step 2.3.8.5.4

Rewrite the expression.

$π \frac{3}{2}$

Step 2.3.8.6

Combine $π$ and $\frac{3}{2}$ .

$\frac{π \cdot 3}{2}$

Step 2.3.8.7

Move $3$ to the left of $π$ .

$\frac{3 π}{2}$

Step 2.3.9

The period of the $\sin (\frac{4 x}{3})$ function is $\frac{3 π}{2}$ so values will repeat every $\frac{3 π}{2}$ radians in both directions.

$x = \frac{3 π n}{2}, \frac{3 π}{4} + \frac{3 π n}{2}$ , for any integer $n$

Step 2.4

Consolidate the answers.

$x = \frac{3 π n}{4}$ , for any integer $n$

Step 3

The values which make the derivative equal to $0$ are $\frac{3 π n}{4}$ .

$\frac{3 π n}{4}$

Step 4

After finding the point that makes the derivative $f' (x) = - \frac{4 \sin (\frac{4 x}{3})}{3}$ equal to $0$ or undefined, the interval to check where $f (x) = \cos (\frac{4 x}{3})$ is increasing and where it is decreasing is $(- \infty, \frac{3 π n}{4}) \cup (\frac{3 π n}{4}, \infty)$ .

$(- \infty, \frac{3 π n}{4}) \cup (\frac{3 π n}{4}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{3 π n}{4})$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $\frac{3 π n}{4} - 1$ in the expression.

$f' (\frac{3 π n}{4} - 1) = - \frac{4 \sin (\frac{4 (\frac{3 π n}{4} - 1)}{3})}{3}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f' (\frac{3 π n}{4} - 1) = - \frac{4 \sin (\frac{4 (\frac{3 π n}{4} - 1 \cdot \frac{4}{4})}{3})}{3}$

Step 5.2.1.2

Combine $- 1$ and $\frac{4}{4}$ .

$f' (\frac{3 π n}{4} - 1) = - \frac{4 \sin (\frac{4 (\frac{3 π n}{4} + \frac{- 1 \cdot 4}{4})}{3})}{3}$

Step 5.2.1.3

Combine the numerators over the common denominator.

$f' (\frac{3 π n}{4} - 1) = - \frac{4 \sin (\frac{4 (\frac{3 π n - 1 \cdot 4}{4})}{3})}{3}$

Step 5.2.1.4

Multiply $- 1$ by $4$ .

$f' (\frac{3 π n}{4} - 1) = - \frac{4 \sin (\frac{4 (\frac{3 π n - 4}{4})}{3})}{3}$

Step 5.2.2

Combine $4$ and $\frac{3 π n - 4}{4}$ .

$f' (\frac{3 π n}{4} - 1) = - \frac{4 \sin (\frac{\frac{4 (3 π n - 4)}{4}}{3})}{3}$

Step 5.2.3

Reduce the expression by cancelling the common factors.

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Step 5.2.3.1

Reduce the expression $\frac{4 (3 π n - 4)}{4}$ by cancelling the common factors.

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Step 5.2.3.1.1

Cancel the common factor.

$f' (\frac{3 π n}{4} - 1) = - \frac{4 \sin (\frac{\frac{4 (3 π n - 4)}{4}}{3})}{3}$

Step 5.2.3.1.2

Rewrite the expression.

$f' (\frac{3 π n}{4} - 1) = - \frac{4 \sin (\frac{\frac{3 π n - 4}{1}}{3})}{3}$

Step 5.2.3.2

Divide $3 π n - 4$ by $1$ .

$f' (\frac{3 π n}{4} - 1) = - \frac{4 \sin (\frac{3 π n - 4}{3})}{3}$

Step 5.2.4

The final answer is $- \frac{4 \sin (\frac{3 π n - 4}{3})}{3}$ .

$- \frac{4 \sin (\frac{3 π n - 4}{3})}{3}$

Step 5.3

At $x = \frac{3 π n}{4} - 1$ the derivative is $- \frac{4 \sin (\frac{3 π n - 4}{3})}{3}$ . Since this is negative, the function is decreasing on $(- \infty, \frac{3 π n}{4})$ .

Decreasing on $(- \infty, \frac{3 π n}{4})$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(\frac{3 π n}{4}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $\frac{3 π n}{4} + 1$ in the expression.

$f' (\frac{3 π n}{4} + 1) = - \frac{4 \sin (\frac{4 (\frac{3 π n}{4} + 1)}{3})}{3}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

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

$f' (\frac{3 π n}{4} + 1) = - \frac{4 \sin (\frac{4 (\frac{3 π n}{4} + \frac{4}{4})}{3})}{3}$

Step 6.2.1.2

Combine the numerators over the common denominator.

$f' (\frac{3 π n}{4} + 1) = - \frac{4 \sin (\frac{4 (\frac{3 π n + 4}{4})}{3})}{3}$

Step 6.2.2

Combine $4$ and $\frac{3 π n + 4}{4}$ .

$f' (\frac{3 π n}{4} + 1) = - \frac{4 \sin (\frac{\frac{4 (3 π n + 4)}{4}}{3})}{3}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

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Step 6.2.3.1

Reduce the expression $\frac{4 (3 π n + 4)}{4}$ by cancelling the common factors.

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Step 6.2.3.1.1

Cancel the common factor.

$f' (\frac{3 π n}{4} + 1) = - \frac{4 \sin (\frac{\frac{4 (3 π n + 4)}{4}}{3})}{3}$

Step 6.2.3.1.2

Rewrite the expression.

$f' (\frac{3 π n}{4} + 1) = - \frac{4 \sin (\frac{\frac{3 π n + 4}{1}}{3})}{3}$

Step 6.2.3.2

Divide $3 π n + 4$ by $1$ .

$f' (\frac{3 π n}{4} + 1) = - \frac{4 \sin (\frac{3 π n + 4}{3})}{3}$

Step 6.2.4

The final answer is $- \frac{4 \sin (\frac{3 π n + 4}{3})}{3}$ .

$- \frac{4 \sin (\frac{3 π n + 4}{3})}{3}$

Step 6.3

At $x = \frac{3 π n}{4} + 1$ the derivative is $- \frac{4 \sin (\frac{3 π n + 4}{3})}{3}$ . Since this is negative, the function is decreasing on $(\frac{3 π n}{4}, \infty)$ .

Decreasing on $(\frac{3 π n}{4}, \infty)$ since $f' (x) < 0$

Step 7

List the intervals on which the function is increasing and decreasing.

Decreasing on: $(- \infty, \frac{3 π n}{4}), (\frac{3 π n}{4}, \infty)$

Step 8