Calculus Examples

$6 x + \sin (6 x)$

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

By the Sum Rule, the derivative of $6 x + \sin (6 x)$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [\sin (6 x)]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [\sin (6 x)]$

Step 1.1.2

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 1.1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x] + \frac{d}{d x} [\sin (6 x)]$

Step 1.1.2.2

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

$6 \cdot 1 + \frac{d}{d x} [\sin (6 x)]$

Step 1.1.2.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [\sin (6 x)]$

Step 1.1.3

Evaluate $\frac{d}{d x} [\sin (6 x)]$ .

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Step 1.1.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 6 x$ .

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Step 1.1.3.1.1

To apply the Chain Rule, set $u$ as $6 x$ .

$6 + \frac{d}{d u} [\sin (u)] \frac{d}{d x} [6 x]$

Step 1.1.3.1.2

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

$6 + \cos (u) \frac{d}{d x} [6 x]$

Step 1.1.3.1.3

Replace all occurrences of $u$ with $6 x$ .

$6 + \cos (6 x) \frac{d}{d x} [6 x]$

Step 1.1.3.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 + \cos (6 x) (6 \frac{d}{d x} [x])$

Step 1.1.3.3

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

$6 + \cos (6 x) (6 \cdot 1)$

Step 1.1.3.4

Multiply $6$ by $1$ .

$6 + \cos (6 x) \cdot 6$

Step 1.1.3.5

Move $6$ to the left of $\cos (6 x)$ .

$f' (x) = 6 + 6 \cos (6 x)$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $6 + 6 \cos (6 x)$ .

$6 + 6 \cos (6 x)$

Step 2

Set the first derivative equal to $0$ then solve the equation $6 + 6 \cos (6 x) = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$6 + 6 \cos (6 x) = 0$

Step 2.2

Subtract $6$ from both sides of the equation.

$6 \cos (6 x) = - 6$

Step 2.3

Divide each term in $6 \cos (6 x) = - 6$ by $6$ and simplify.

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Step 2.3.1

Divide each term in $6 \cos (6 x) = - 6$ by $6$ .

$\frac{6 \cos (6 x)}{6} = \frac{- 6}{6}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $6$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{6 \cos (6 x)}{6} = \frac{- 6}{6}$

Step 2.3.2.1.2

Divide $\cos (6 x)$ by $1$ .

$\cos (6 x) = \frac{- 6}{6}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 6$ by $6$ .

$\cos (6 x) = - 1$

Step 2.4

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

$6 x = \arccos (- 1)$

Step 2.5

Simplify the right side.

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Step 2.5.1

The exact value of $\arccos (- 1)$ is $π$ .

$6 x = π$

Step 2.6

Divide each term in $6 x = π$ by $6$ and simplify.

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Step 2.6.1

Divide each term in $6 x = π$ by $6$ .

$\frac{6 x}{6} = \frac{π}{6}$

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Cancel the common factor of $6$ .

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Step 2.6.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{π}{6}$

Step 2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{6}$

Step 2.7

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$6 x = 2 π - π$

Step 2.8

Solve for $x$ .

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Step 2.8.1

Subtract $π$ from $2 π$ .

$6 x = π$

Step 2.8.2

Divide each term in $6 x = π$ by $6$ and simplify.

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Step 2.8.2.1

Divide each term in $6 x = π$ by $6$ .

$\frac{6 x}{6} = \frac{π}{6}$

Step 2.8.2.2

Simplify the left side.

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Step 2.8.2.2.1

Cancel the common factor of $6$ .

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Step 2.8.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{π}{6}$

Step 2.8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{6}$

Step 2.9

Find the period of $\cos (6 x)$ .

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Step 2.9.1

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

$\frac{2 π}{| b |}$

Step 2.9.2

Replace $b$ with $6$ in the formula for period.

$\frac{2 π}{| 6 |}$

Step 2.9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$\frac{2 π}{6}$

Step 2.9.4

Cancel the common factor of $2$ and $6$ .

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Step 2.9.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{6}$

Step 2.9.4.2

Cancel the common factors.

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Step 2.9.4.2.1

Factor $2$ out of $6$ .

$\frac{2 π}{2 \cdot 3}$

Step 2.9.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 3}$

Step 2.9.4.2.3

Rewrite the expression.

$\frac{π}{3}$

Step 2.10

The period of the $\cos (6 x)$ function is $\frac{π}{3}$ so values will repeat every $\frac{π}{3}$ radians in both directions.

$x = \frac{π}{6} + \frac{π n}{3}$ , 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 $6 x + \sin (6 x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{π}{6}$ .

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Step 4.1.1

Substitute $\frac{π}{6}$ for $x$ .

$6 (\frac{π}{6}) + \sin (6 (\frac{π}{6}))$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Cancel the common factor of $6$ .

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Step 4.1.2.1.1.1

Cancel the common factor.

$6 \frac{π}{6} + \sin (6 (\frac{π}{6}))$

Step 4.1.2.1.1.2

Rewrite the expression.

$π + \sin (6 (\frac{π}{6}))$

Step 4.1.2.1.2

Cancel the common factor of $6$ .

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Step 4.1.2.1.2.1

Cancel the common factor.

$π + \sin (6 \frac{π}{6})$

Step 4.1.2.1.2.2

Rewrite the expression.

$π + \sin (π)$

Step 4.1.2.1.3

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

$π + \sin (0)$

Step 4.1.2.1.4

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

$π + 0$

Step 4.1.2.2

Add $π$ and $0$ .

$π$

Step 4.2

Evaluate at $x = \frac{π}{2}$ .

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Step 4.2.1

Substitute $\frac{π}{2}$ for $x$ .

$6 (\frac{π}{2}) + \sin (6 (\frac{π}{2}))$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.1

Factor $2$ out of $6$ .

$2 (3) \frac{π}{2} + \sin (6 (\frac{π}{2}))$

Step 4.2.2.1.1.2

Cancel the common factor.

$2 \cdot 3 \frac{π}{2} + \sin (6 (\frac{π}{2}))$

Step 4.2.2.1.1.3

Rewrite the expression.

$3 π + \sin (6 (\frac{π}{2}))$

Step 4.2.2.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.2.1

Factor $2$ out of $6$ .

$3 π + \sin (2 (3) \frac{π}{2})$

Step 4.2.2.1.2.2

Cancel the common factor.

$3 π + \sin (2 \cdot 3 \frac{π}{2})$

Step 4.2.2.1.2.3

Rewrite the expression.

$3 π + \sin (3 π)$

Step 4.2.2.1.3

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$3 π + \sin (π)$

Step 4.2.2.1.4

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

$3 π + \sin (0)$

Step 4.2.2.1.5

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

$3 π + 0$

Step 4.2.2.2

Add $3 π$ and $0$ .

$3 π$

Step 4.3

Evaluate at $x = \frac{5 π}{6}$ .

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Step 4.3.1

Substitute $\frac{5 π}{6}$ for $x$ .

$6 (\frac{5 π}{6}) + \sin (6 (\frac{5 π}{6}))$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Cancel the common factor of $6$ .

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Step 4.3.2.1.1.1

Cancel the common factor.

$6 \frac{5 π}{6} + \sin (6 (\frac{5 π}{6}))$

Step 4.3.2.1.1.2

Rewrite the expression.

$5 π + \sin (6 (\frac{5 π}{6}))$

Step 4.3.2.1.2

Cancel the common factor of $6$ .

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Step 4.3.2.1.2.1

Cancel the common factor.

$5 π + \sin (6 \frac{5 π}{6})$

Step 4.3.2.1.2.2

Rewrite the expression.

$5 π + \sin (5 π)$

Step 4.3.2.1.3

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$5 π + \sin (π)$

Step 4.3.2.1.4

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

$5 π + \sin (0)$

Step 4.3.2.1.5

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

$5 π + 0$

Step 4.3.2.2

Add $5 π$ and $0$ .

$5 π$

Step 4.4

Evaluate at $x = \frac{7 π}{6}$ .

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Step 4.4.1

Substitute $\frac{7 π}{6}$ for $x$ .

$6 (\frac{7 π}{6}) + \sin (6 (\frac{7 π}{6}))$

Step 4.4.2

Simplify.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Cancel the common factor of $6$ .

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Step 4.4.2.1.1.1

Cancel the common factor.

$6 \frac{7 π}{6} + \sin (6 (\frac{7 π}{6}))$

Step 4.4.2.1.1.2

Rewrite the expression.

$7 π + \sin (6 (\frac{7 π}{6}))$

Step 4.4.2.1.2

Cancel the common factor of $6$ .

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Step 4.4.2.1.2.1

Cancel the common factor.

$7 π + \sin (6 \frac{7 π}{6})$

Step 4.4.2.1.2.2

Rewrite the expression.

$7 π + \sin (7 π)$

Step 4.4.2.1.3

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$7 π + \sin (π)$

Step 4.4.2.1.4

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

$7 π + \sin (0)$

Step 4.4.2.1.5

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

$7 π + 0$

Step 4.4.2.2

Add $7 π$ and $0$ .

$7 π$

Step 4.5

Evaluate at $x = \frac{3 π}{2}$ .

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Step 4.5.1

Substitute $\frac{3 π}{2}$ for $x$ .

$6 (\frac{3 π}{2}) + \sin (6 (\frac{3 π}{2}))$

Step 4.5.2

Simplify.

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Step 4.5.2.1

Simplify each term.

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Step 4.5.2.1.1

Cancel the common factor of $2$ .

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Step 4.5.2.1.1.1

Factor $2$ out of $6$ .

$2 (3) \frac{3 π}{2} + \sin (6 (\frac{3 π}{2}))$

Step 4.5.2.1.1.2

Cancel the common factor.

$2 \cdot 3 \frac{3 π}{2} + \sin (6 (\frac{3 π}{2}))$

Step 4.5.2.1.1.3

Rewrite the expression.

$3 (3 π) + \sin (6 (\frac{3 π}{2}))$

Step 4.5.2.1.2

Multiply $3$ by $3$ .

$9 π + \sin (6 (\frac{3 π}{2}))$

Step 4.5.2.1.3

Cancel the common factor of $2$ .

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Step 4.5.2.1.3.1

Factor $2$ out of $6$ .

$9 π + \sin (2 (3) \frac{3 π}{2})$

Step 4.5.2.1.3.2

Cancel the common factor.

$9 π + \sin (2 \cdot 3 \frac{3 π}{2})$

Step 4.5.2.1.3.3

Rewrite the expression.

$9 π + \sin (3 (3 π))$

Step 4.5.2.1.4

Multiply $3$ by $3$ .

$9 π + \sin (9 π)$

Step 4.5.2.1.5

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$9 π + \sin (π)$

Step 4.5.2.1.6

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

$9 π + \sin (0)$

Step 4.5.2.1.7

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

$9 π + 0$

Step 4.5.2.2

Add $9 π$ and $0$ .

$9 π$

Step 4.6

List all of the points.

$(\frac{π}{6}, π), (\frac{π}{2}, 3 π), (\frac{5 π}{6}, 5 π), (\frac{7 π}{6}, 7 π), (\frac{3 π}{2}, 9 π)$

Step 5