Calculus Examples

$f (x) = \cos (4 x)$

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 $g (x) = 4 x$ .

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Step 1.1.1.1

To apply the Chain Rule, set $u$ as $4 x$ .

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

Step 1.1.1.2

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

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

Step 1.1.1.3

Replace all occurrences of $u$ with $4 x$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

Multiply $4$ by $- 1$ .

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

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 $n = 1$ .

$- 4 \sin (4 x) \cdot 1$

Step 1.1.2.4

Multiply $- 4$ by $1$ .

$f' (x) = - 4 \sin (4 x)$

Step 1.2

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

$- 4 \sin (4 x)$

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

$- 4 \sin (4 x) = 0$

Step 2.2

Divide each term in $- 4 \sin (4 x) = 0$ by $- 4$ and simplify.

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Step 2.2.1

Divide each term in $- 4 \sin (4 x) = 0$ by $- 4$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 4$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $\sin (4 x)$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $0$ by $- 4$ .

$\sin (4 x) = 0$

Step 2.3

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

$4 x = \arcsin (0)$

Step 2.4

Simplify the right side.

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Step 2.4.1

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

$4 x = 0$

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $4$ .

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Step 2.5.2.1.1

Cancel the common factor.

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

Step 2.5.2.1.2

Divide $x$ by $1$ .

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

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Divide $0$ by $4$ .

$x = 0$

Step 2.6

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.

$4 x = π - 0$

Step 2.7

Solve for $x$ .

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Step 2.7.1

Simplify.

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Step 2.7.1.1

Multiply $- 1$ by $0$ .

$4 x = π + 0$

Step 2.7.1.2

Add $π$ and $0$ .

$4 x = π$

Step 2.7.2

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

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Step 2.7.2.1

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

$\frac{4 x}{4} = \frac{π}{4}$

Step 2.7.2.2

Simplify the left side.

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Step 2.7.2.2.1

Cancel the common factor of $4$ .

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Step 2.7.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{π}{4}$

Step 2.7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{4}$

Step 2.8

Find the period of $\sin (4 x)$ .

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Step 2.8.1

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

$\frac{2 π}{| b |}$

Step 2.8.2

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Step 2.8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Step 2.8.4

Cancel the common factor of $2$ and $4$ .

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Step 2.8.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 2.8.4.2

Cancel the common factors.

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Step 2.8.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 2.8.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 2.8.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 2.9

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

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

Step 2.10

Consolidate the answers.

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

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $\cos (4 x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$\cos (4 (0))$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $4$ by $0$ .

$\cos (0)$

Step 4.1.2.2

The exact value of $\cos (0)$ is $1$ .

$1$

Step 4.2

Evaluate at $x = \frac{π}{4}$ .

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Step 4.2.1

Substitute $\frac{π}{4}$ for $x$ .

$\cos (4 (\frac{π}{4}))$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\cos (4 \frac{π}{4})$

Step 4.2.2.1.2

Rewrite the expression.

$\cos (π)$

Step 4.2.2.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$- \cos (0)$

Step 4.2.2.3

The exact value of $\cos (0)$ is $1$ .

$- 1 \cdot 1$

Step 4.2.2.4

Multiply $- 1$ by $1$ .

$- 1$

Step 4.3

List all of the points.

$(0 + \frac{π n}{2}, 1), (\frac{π}{4} + \frac{π n}{2}, - 1)$ , for any integer $n$

Step 5