Calculus Examples

$y = 6 x - 4 \cos (3 x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (6 x - 4 \cos (3 x))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $6 x - 4 \cos (3 x)$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- 4 \cos (3 x)]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [- 4 \cos (3 x)]$

Step 3.2

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

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Step 3.2.1

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

$6 \frac{d}{d x} [x] + \frac{d}{d x} [- 4 \cos (3 x)]$

Step 3.2.2

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

$6 \cdot 1 + \frac{d}{d x} [- 4 \cos (3 x)]$

Step 3.2.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [- 4 \cos (3 x)]$

Step 3.3

Evaluate $\frac{d}{d x} [- 4 \cos (3 x)]$ .

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Step 3.3.1

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

$6 - 4 \frac{d}{d x} [\cos (3 x)]$

Step 3.3.2

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 3.3.2.1

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

$6 - 4 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [3 x])$

Step 3.3.2.2

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

$6 - 4 (- \sin (u) \frac{d}{d x} [3 x])$

Step 3.3.2.3

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

$6 - 4 (- \sin (3 x) \frac{d}{d x} [3 x])$

Step 3.3.3

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

$6 - 4 (- \sin (3 x) (3 \frac{d}{d x} [x]))$

Step 3.3.4

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

$6 - 4 (- \sin (3 x) (3 \cdot 1))$

Step 3.3.5

Multiply $3$ by $1$ .

$6 - 4 (- \sin (3 x) \cdot 3)$

Step 3.3.6

Multiply $3$ by $- 1$ .

$6 - 4 (- 3 \sin (3 x))$

Step 3.3.7

Multiply $- 3$ by $- 4$ .

$6 + 12 \sin (3 x)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 6 + 12 \sin (3 x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 6 + 12 \sin (3 x)$

Step 6

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Step 6.1

Subtract $6$ from both sides of the equation.

$12 \sin (3 x) = - 6$

Step 6.2

Divide each term in $12 \sin (3 x) = - 6$ by $12$ and simplify.

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Step 6.2.1

Divide each term in $12 \sin (3 x) = - 6$ by $12$ .

$\frac{12 \sin (3 x)}{12} = \frac{- 6}{12}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $12$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{12 \sin (3 x)}{12} = \frac{- 6}{12}$

Step 6.2.2.1.2

Divide $\sin (3 x)$ by $1$ .

$\sin (3 x) = \frac{- 6}{12}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Cancel the common factor of $- 6$ and $12$ .

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Step 6.2.3.1.1

Factor $6$ out of $- 6$ .

$\sin (3 x) = \frac{6 (- 1)}{12}$

Step 6.2.3.1.2

Cancel the common factors.

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Step 6.2.3.1.2.1

Factor $6$ out of $12$ .

$\sin (3 x) = \frac{6 \cdot - 1}{6 \cdot 2}$

Step 6.2.3.1.2.2

Cancel the common factor.

$\sin (3 x) = \frac{6 \cdot - 1}{6 \cdot 2}$

Step 6.2.3.1.2.3

Rewrite the expression.

$\sin (3 x) = \frac{- 1}{2}$

Step 6.2.3.2

Move the negative in front of the fraction.

$\sin (3 x) = - \frac{1}{2}$

Step 6.3

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

$3 x = \arcsin (- \frac{1}{2})$

Step 6.4

Simplify the right side.

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Step 6.4.1

The exact value of $\arcsin (- \frac{1}{2})$ is $- \frac{π}{6}$ .

$3 x = - \frac{π}{6}$

Step 6.5

Divide each term in $3 x = - \frac{π}{6}$ by $3$ and simplify.

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Step 6.5.1

Divide each term in $3 x = - \frac{π}{6}$ by $3$ .

$\frac{3 x}{3} = \frac{- \frac{π}{6}}{3}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $3$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- \frac{π}{6}}{3}$

Step 6.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{6}}{3}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{6} \cdot \frac{1}{3}$

Step 6.5.3.2

Multiply $- \frac{π}{6} \cdot \frac{1}{3}$ .

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Step 6.5.3.2.1

Multiply $\frac{1}{3}$ by $\frac{π}{6}$ .

$x = - \frac{π}{3 \cdot 6}$

Step 6.5.3.2.2

Multiply $3$ by $6$ .

$x = - \frac{π}{18}$

Step 6.6

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$3 x = 2 π + \frac{π}{6} + π$

Step 6.7

Simplify the expression to find the second solution.

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Step 6.7.1

Subtract $2 π$ from $2 π + \frac{π}{6} + π$ .

$3 x = 2 π + \frac{π}{6} + π - 2 π$

Step 6.7.2

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{6} + π$ .

$3 x = \frac{7 π}{6}$

Step 6.7.3

Divide each term in $3 x = \frac{7 π}{6}$ by $3$ and simplify.

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Step 6.7.3.1

Divide each term in $3 x = \frac{7 π}{6}$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{7 π}{6}}{3}$

Step 6.7.3.2

Simplify the left side.

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Step 6.7.3.2.1

Cancel the common factor of $3$ .

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Step 6.7.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{7 π}{6}}{3}$

Step 6.7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{7 π}{6}}{3}$

Step 6.7.3.3

Simplify the right side.

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Step 6.7.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7 π}{6} \cdot \frac{1}{3}$

Step 6.7.3.3.2

Multiply $\frac{7 π}{6} \cdot \frac{1}{3}$ .

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Step 6.7.3.3.2.1

Multiply $\frac{7 π}{6}$ by $\frac{1}{3}$ .

$x = \frac{7 π}{6 \cdot 3}$

Step 6.7.3.3.2.2

Multiply $6$ by $3$ .

$x = \frac{7 π}{18}$

Step 6.8

Find the period of $\sin (3 x)$ .

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Step 6.8.1

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

$\frac{2 π}{| b |}$

Step 6.8.2

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 6.8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Step 6.9

Add $\frac{2 π}{3}$ to every negative angle to get positive angles.

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Step 6.9.1

Add $\frac{2 π}{3}$ to $- \frac{π}{18}$ to find the positive angle.

$- \frac{π}{18} + \frac{2 π}{3}$

Step 6.9.2

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{2 π}{3} \cdot \frac{6}{6} - \frac{π}{18}$

Step 6.9.3

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.9.3.1

Multiply $\frac{2 π}{3}$ by $\frac{6}{6}$ .

$\frac{2 π \cdot 6}{3 \cdot 6} - \frac{π}{18}$

Step 6.9.3.2

Multiply $3$ by $6$ .

$\frac{2 π \cdot 6}{18} - \frac{π}{18}$

Step 6.9.4

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{18}$

Step 6.9.5

Simplify the numerator.

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Step 6.9.5.1

Multiply $6$ by $2$ .

$\frac{12 π - π}{18}$

Step 6.9.5.2

Subtract $π$ from $12 π$ .

$\frac{11 π}{18}$

Step 6.9.6

List the new angles.

$x = \frac{11 π}{18}$

Step 6.10

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

$x = \frac{7 π}{18} + \frac{2 π n}{3}, \frac{11 π}{18} + \frac{2 π n}{3}$ , for any integer $n$

Step 7

Simplify $6 (\frac{7 π}{18} + \frac{2 π n}{3}) - 4 \cos (3 (\frac{7 π}{18} + \frac{2 π n}{3}))$ .

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Step 7.1

Simplify each term.

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Step 7.1.1

Apply the distributive property.

$y = 6 \frac{7 π}{18} + 6 \frac{2 π n}{3} - 4 \cos (3 (\frac{7 π}{18} + \frac{2 π n}{3}))$

Step 7.1.2

Cancel the common factor of $6$ .

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Step 7.1.2.1

Factor $6$ out of $18$ .

$y = 6 \frac{7 π}{6 (3)} + 6 \frac{2 π n}{3} - 4 \cos (3 (\frac{7 π}{18} + \frac{2 π n}{3}))$

Step 7.1.2.2

Cancel the common factor.

$y = 6 \frac{7 π}{6 \cdot 3} + 6 \frac{2 π n}{3} - 4 \cos (3 (\frac{7 π}{18} + \frac{2 π n}{3}))$

Step 7.1.2.3

Rewrite the expression.

$y = \frac{7 π}{3} + 6 \frac{2 π n}{3} - 4 \cos (3 (\frac{7 π}{18} + \frac{2 π n}{3}))$

Step 7.1.3

Cancel the common factor of $3$ .

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Step 7.1.3.1

Factor $3$ out of $6$ .

$y = \frac{7 π}{3} + 3 (2) \frac{2 π n}{3} - 4 \cos (3 (\frac{7 π}{18} + \frac{2 π n}{3}))$

Step 7.1.3.2

Cancel the common factor.

$y = \frac{7 π}{3} + 3 \cdot 2 \frac{2 π n}{3} - 4 \cos (3 (\frac{7 π}{18} + \frac{2 π n}{3}))$

Step 7.1.3.3

Rewrite the expression.

$y = \frac{7 π}{3} + 2 (2 π n) - 4 \cos (3 (\frac{7 π}{18} + \frac{2 π n}{3}))$

Step 7.1.4

Multiply $2$ by $2$ .

$y = \frac{7 π}{3} + 4 (π n) - 4 \cos (3 (\frac{7 π}{18} + \frac{2 π n}{3}))$

Step 7.1.5

Apply the distributive property.

$y = \frac{7 π}{3} + 4 π n - 4 \cos (3 \frac{7 π}{18} + 3 \frac{2 π n}{3})$

Step 7.1.6

Cancel the common factor of $3$ .

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Step 7.1.6.1

Factor $3$ out of $18$ .

$y = \frac{7 π}{3} + 4 π n - 4 \cos (3 \frac{7 π}{3 (6)} + 3 \frac{2 π n}{3})$

Step 7.1.6.2

Cancel the common factor.

$y = \frac{7 π}{3} + 4 π n - 4 \cos (3 \frac{7 π}{3 \cdot 6} + 3 \frac{2 π n}{3})$

Step 7.1.6.3

Rewrite the expression.

$y = \frac{7 π}{3} + 4 π n - 4 \cos (\frac{7 π}{6} + 3 \frac{2 π n}{3})$

Step 7.1.7

Cancel the common factor of $3$ .

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Step 7.1.7.1

Cancel the common factor.

$y = \frac{7 π}{3} + 4 π n - 4 \cos (\frac{7 π}{6} + 3 \frac{2 π n}{3})$

Step 7.1.7.2

Rewrite the expression.

$y = \frac{7 π}{3} + 4 π n - 4 \cos (\frac{7 π}{6} + 2 π n)$

Step 7.2

Simplify with commuting.

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Step 7.2.1

Reorder $\frac{7 π}{6}$ and $2 π n$ .

$y = \frac{7 π}{3} + 4 π n - 4 \cos (2 π n + \frac{7 π}{6})$

Step 7.2.2

Reorder $\frac{7 π}{3}$ and $4 π n$ .

$y = 4 π n + \frac{7 π}{3} - 4 \cos (2 π n + \frac{7 π}{6})$

Step 8

Simplify $6 (\frac{11 π}{18} + \frac{2 π n}{3}) - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$ .

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Step 8.1

Simplify each term.

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Step 8.1.1

Apply the distributive property.

$y = 6 \frac{11 π}{18} + 6 \frac{2 π n}{3} - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Step 8.1.2

Cancel the common factor of $6$ .

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Step 8.1.2.1

Factor $6$ out of $18$ .

$y = 6 \frac{11 π}{6 (3)} + 6 \frac{2 π n}{3} - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Step 8.1.2.2

Cancel the common factor.

$y = 6 \frac{11 π}{6 \cdot 3} + 6 \frac{2 π n}{3} - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Step 8.1.2.3

Rewrite the expression.

$y = \frac{11 π}{3} + 6 \frac{2 π n}{3} - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Step 8.1.3

Cancel the common factor of $3$ .

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Step 8.1.3.1

Factor $3$ out of $6$ .

$y = \frac{11 π}{3} + 3 (2) \frac{2 π n}{3} - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Step 8.1.3.2

Cancel the common factor.

$y = \frac{11 π}{3} + 3 \cdot 2 \frac{2 π n}{3} - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Step 8.1.3.3

Rewrite the expression.

$y = \frac{11 π}{3} + 2 (2 π n) - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Step 8.1.4

Multiply $2$ by $2$ .

$y = \frac{11 π}{3} + 4 (π n) - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Step 8.1.5

Apply the distributive property.

$y = \frac{11 π}{3} + 4 π n - 4 \cos (3 \frac{11 π}{18} + 3 \frac{2 π n}{3})$

Step 8.1.6

Cancel the common factor of $3$ .

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Step 8.1.6.1

Factor $3$ out of $18$ .

$y = \frac{11 π}{3} + 4 π n - 4 \cos (3 \frac{11 π}{3 (6)} + 3 \frac{2 π n}{3})$

Step 8.1.6.2

Cancel the common factor.

$y = \frac{11 π}{3} + 4 π n - 4 \cos (3 \frac{11 π}{3 \cdot 6} + 3 \frac{2 π n}{3})$

Step 8.1.6.3

Rewrite the expression.

$y = \frac{11 π}{3} + 4 π n - 4 \cos (\frac{11 π}{6} + 3 \frac{2 π n}{3})$

Step 8.1.7

Cancel the common factor of $3$ .

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Step 8.1.7.1

Cancel the common factor.

$y = \frac{11 π}{3} + 4 π n - 4 \cos (\frac{11 π}{6} + 3 \frac{2 π n}{3})$

Step 8.1.7.2

Rewrite the expression.

$y = \frac{11 π}{3} + 4 π n - 4 \cos (\frac{11 π}{6} + 2 π n)$

Step 8.2

Simplify with commuting.

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Step 8.2.1

Reorder $\frac{11 π}{6}$ and $2 π n$ .

$y = \frac{11 π}{3} + 4 π n - 4 \cos (2 π n + \frac{11 π}{6})$

Step 8.2.2

Reorder $\frac{11 π}{3}$ and $4 π n$ .

$y = 4 π n + \frac{11 π}{3} - 4 \cos (2 π n + \frac{11 π}{6})$

Step 9

Find the points where $\frac{d y}{d x} = 0$ .

$(\frac{7 π}{18} + \frac{2 π n}{3}, 4 π n + \frac{7 π}{3} - 4 \cos (2 π n + \frac{7 π}{6}))$ , for any integer $n$

$(\frac{11 π}{18} + \frac{2 π n}{3}, 4 π n + \frac{11 π}{3} - 4 \cos (2 π n + \frac{11 π}{6}))$ , for any integer $n$

Step 10