Calculus Examples

$\frac{\cos (x)}{2 + \sin (x)}$

Step 1

Find the derivative.

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Step 1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \cos (x)$ and $g (x) = 2 + \sin (x)$ .

$\frac{(2 + \sin (x)) \frac{d}{d x} [\cos (x)] - \cos (x) \frac{d}{d x} [2 + \sin (x)]}{{(2 + \sin (x))}^{2}}$

Step 1.2

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

$\frac{(2 + \sin (x)) (- \sin (x)) - \cos (x) \frac{d}{d x} [2 + \sin (x)]}{{(2 + \sin (x))}^{2}}$

Step 1.3

Differentiate.

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Step 1.3.1

By the Sum Rule, the derivative of $2 + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [\sin (x)]$ .

$\frac{(2 + \sin (x)) (- \sin (x)) - \cos (x) (\frac{d}{d x} [2] + \frac{d}{d x} [\sin (x)])}{{(2 + \sin (x))}^{2}}$

Step 1.3.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{(2 + \sin (x)) (- \sin (x)) - \cos (x) (0 + \frac{d}{d x} [\sin (x)])}{{(2 + \sin (x))}^{2}}$

Step 1.3.3

Add $0$ and $\frac{d}{d x} [\sin (x)]$ .

$\frac{(2 + \sin (x)) (- \sin (x)) - \cos (x) \frac{d}{d x} [\sin (x)]}{{(2 + \sin (x))}^{2}}$

Step 1.4

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

$\frac{(2 + \sin (x)) (- \sin (x)) - \cos (x) \cos (x)}{{(2 + \sin (x))}^{2}}$

Step 1.5

Raise $\cos (x)$ to the power of $1$ .

$\frac{(2 + \sin (x)) (- \sin (x)) - (\cos^{1} (x) \cos (x))}{{(2 + \sin (x))}^{2}}$

Step 1.6

Raise $\cos (x)$ to the power of $1$ .

$\frac{(2 + \sin (x)) (- \sin (x)) - (\cos^{1} (x) \cos^{1} (x))}{{(2 + \sin (x))}^{2}}$

Step 1.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(2 + \sin (x)) (- \sin (x)) - {\cos (x)}^{1 + 1}}{{(2 + \sin (x))}^{2}}$

Step 1.8

Add $1$ and $1$ .

$\frac{(2 + \sin (x)) (- \sin (x)) - \cos^{2} (x)}{{(2 + \sin (x))}^{2}}$

Step 1.9

Simplify.

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Step 1.9.1

Apply the distributive property.

$\frac{2 (- \sin (x)) + \sin (x) (- \sin (x)) - \cos^{2} (x)}{{(2 + \sin (x))}^{2}}$

Step 1.9.2

Simplify the numerator.

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Step 1.9.2.1

Simplify each term.

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Step 1.9.2.1.1

Multiply $- 1$ by $2$ .

$\frac{- 2 \sin (x) + \sin (x) (- \sin (x)) - \cos^{2} (x)}{{(2 + \sin (x))}^{2}}$

Step 1.9.2.1.2

Rewrite using the commutative property of multiplication.

$\frac{- 2 \sin (x) - \sin (x) \sin (x) - \cos^{2} (x)}{{(2 + \sin (x))}^{2}}$

Step 1.9.2.1.3

Multiply $- \sin (x) \sin (x)$ .

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Step 1.9.2.1.3.1

Raise $\sin (x)$ to the power of $1$ .

$\frac{- 2 \sin (x) - (\sin^{1} (x) \sin (x)) - \cos^{2} (x)}{{(2 + \sin (x))}^{2}}$

Step 1.9.2.1.3.2

Raise $\sin (x)$ to the power of $1$ .

$\frac{- 2 \sin (x) - (\sin^{1} (x) \sin^{1} (x)) - \cos^{2} (x)}{{(2 + \sin (x))}^{2}}$

Step 1.9.2.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 2 \sin (x) - {\sin (x)}^{1 + 1} - \cos^{2} (x)}{{(2 + \sin (x))}^{2}}$

Step 1.9.2.1.3.4

Add $1$ and $1$ .

$\frac{- 2 \sin (x) - \sin^{2} (x) - \cos^{2} (x)}{{(2 + \sin (x))}^{2}}$

Step 1.9.2.2

Factor $- 1$ out of $- \sin^{2} (x)$ .

$\frac{- 2 \sin (x) - (\sin^{2} (x)) - \cos^{2} (x)}{{(2 + \sin (x))}^{2}}$

Step 1.9.2.3

Factor $- 1$ out of $- \cos^{2} (x)$ .

$\frac{- 2 \sin (x) - (\sin^{2} (x)) - (\cos^{2} (x))}{{(2 + \sin (x))}^{2}}$

Step 1.9.2.4

Factor $- 1$ out of $- (\sin^{2} (x)) - (\cos^{2} (x))$ .

$\frac{- 2 \sin (x) - (\sin^{2} (x) + \cos^{2} (x))}{{(2 + \sin (x))}^{2}}$

Step 1.9.2.5

Apply pythagorean identity.

$\frac{- 2 \sin (x) - 1 \cdot 1}{{(2 + \sin (x))}^{2}}$

Step 1.9.2.6

Multiply $- 1$ by $1$ .

$\frac{- 2 \sin (x) - 1}{{(2 + \sin (x))}^{2}}$

Step 1.9.3

Reorder terms.

$\frac{- 1 - 2 \sin (x)}{{(2 + \sin (x))}^{2}}$

Step 1.9.4

Factor $- 1$ out of $- 1 - 2 \sin (x)$ .

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Step 1.9.4.1

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- 1 (1) - 2 \sin (x)}{{(2 + \sin (x))}^{2}}$

Step 1.9.4.2

Factor $- 1$ out of $- 2 \sin (x)$ .

$\frac{- 1 (1) - (2 \sin (x))}{{(2 + \sin (x))}^{2}}$

Step 1.9.4.3

Factor $- 1$ out of $- 1 (1) - (2 \sin (x))$ .

$\frac{- 1 (1 + 2 \sin (x))}{{(2 + \sin (x))}^{2}}$

Step 1.9.4.4

Rewrite $- 1 (1 + 2 \sin (x))$ as $- (1 + 2 \sin (x))$ .

$\frac{- (1 + 2 \sin (x))}{{(2 + \sin (x))}^{2}}$

Step 1.9.5

Move the negative in front of the fraction.

$- \frac{1 + 2 \sin (x)}{{(2 + \sin (x))}^{2}}$

Step 2

Set the derivative equal to $0$ then solve the equation $- \frac{1 + 2 \sin (x)}{{(2 + \sin (x))}^{2}} = 0$ .

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Step 2.1

Set the numerator equal to zero.

$1 + 2 \sin (x) = 0$

Step 2.2

Solve the equation for $x$ .

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Step 2.2.1

Subtract $1$ from both sides of the equation.

$2 \sin (x) = - 1$

Step 2.2.2

Divide each term in $2 \sin (x) = - 1$ by $2$ and simplify.

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Step 2.2.2.1

Divide each term in $2 \sin (x) = - 1$ by $2$ .

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

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.2.1.2

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

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

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.2.3

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

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

Step 2.2.4

Simplify the right side.

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Step 2.2.4.1

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

$x = - \frac{π}{6}$

Step 2.2.5

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.

$x = 2 π + \frac{π}{6} + π$

Step 2.2.6

Simplify the expression to find the second solution.

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Step 2.2.6.1

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

$x = 2 π + \frac{π}{6} + π - 2 π$

Step 2.2.6.2

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

$x = \frac{7 π}{6}$

Step 2.2.7

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

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Step 2.2.7.1

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

$\frac{2 π}{| b |}$

Step 2.2.7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 2.2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 2.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.2.8

Add $2 π$ to every negative angle to get positive angles.

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Step 2.2.8.1

Add $2 π$ to $- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + 2 π$

Step 2.2.8.2

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

$2 π \cdot \frac{6}{6} - \frac{π}{6}$

Step 2.2.8.3

Combine fractions.

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Step 2.2.8.3.1

Combine $2 π$ and $\frac{6}{6}$ .

$\frac{2 π \cdot 6}{6} - \frac{π}{6}$

Step 2.2.8.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{6}$

Step 2.2.8.4

Simplify the numerator.

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Step 2.2.8.4.1

Multiply $6$ by $2$ .

$\frac{12 π - π}{6}$

Step 2.2.8.4.2

Subtract $π$ from $12 π$ .

$\frac{11 π}{6}$

Step 2.2.8.5

List the new angles.

$x = \frac{11 π}{6}$

Step 2.2.9

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

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Step 3

Solve the original function $f (x) = \frac{\cos (x)}{2 + \sin (x)}$ at $x = \frac{7 π}{6}$ .

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Step 3.1

Replace the variable $x$ with $\frac{7 π}{6}$ in the expression.

$f (\frac{7 π}{6}) = \frac{\cos (\frac{7 π}{6})}{2 + \sin (\frac{7 π}{6})}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify the numerator.

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Step 3.2.1.1

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 third quadrant.

$f (\frac{7 π}{6}) = \frac{- \cos (\frac{π}{6})}{2 + \sin (\frac{7 π}{6})}$

Step 3.2.1.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$f (\frac{7 π}{6}) = \frac{- \frac{\sqrt{3}}{2}}{2 + \sin (\frac{7 π}{6})}$

Step 3.2.2

Simplify the denominator.

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Step 3.2.2.1

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

$f (\frac{7 π}{6}) = \frac{- \frac{\sqrt{3}}{2}}{2 - \sin (\frac{π}{6})}$

Step 3.2.2.2

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

$f (\frac{7 π}{6}) = \frac{- \frac{\sqrt{3}}{2}}{2 - \frac{1}{2}}$

Step 3.2.2.3

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

$f (\frac{7 π}{6}) = \frac{- \frac{\sqrt{3}}{2}}{2 \cdot \frac{2}{2} - \frac{1}{2}}$

Step 3.2.2.4

Combine $2$ and $\frac{2}{2}$ .

$f (\frac{7 π}{6}) = \frac{- \frac{\sqrt{3}}{2}}{\frac{2 \cdot 2}{2} - \frac{1}{2}}$

Step 3.2.2.5

Combine the numerators over the common denominator.

$f (\frac{7 π}{6}) = \frac{- \frac{\sqrt{3}}{2}}{\frac{2 \cdot 2 - 1}{2}}$

Step 3.2.2.6

Simplify the numerator.

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Step 3.2.2.6.1

Multiply $2$ by $2$ .

$f (\frac{7 π}{6}) = \frac{- \frac{\sqrt{3}}{2}}{\frac{4 - 1}{2}}$

Step 3.2.2.6.2

Subtract $1$ from $4$ .

$f (\frac{7 π}{6}) = \frac{- \frac{\sqrt{3}}{2}}{\frac{3}{2}}$

Step 3.2.3

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7 π}{6}) = - \frac{\sqrt{3}}{2} \cdot \frac{2}{3}$

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$f (\frac{7 π}{6}) = \frac{- \sqrt{3}}{2} \cdot \frac{2}{3}$

Step 3.2.4.2

Cancel the common factor.

$f (\frac{7 π}{6}) = \frac{- \sqrt{3}}{2} \cdot \frac{2}{3}$

Step 3.2.4.3

Rewrite the expression.

$f (\frac{7 π}{6}) = - \sqrt{3} \frac{1}{3}$

Step 3.2.5

Combine $\frac{1}{3}$ and $\sqrt{3}$ .

$f (\frac{7 π}{6}) = - \frac{\sqrt{3}}{3}$

Step 3.2.6

The final answer is $- \frac{\sqrt{3}}{3}$ .

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

Step 4

Solve the original function $f (x) = \frac{\cos (x)}{2 + \sin (x)}$ at $x = \frac{11 π}{6}$ .

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Step 4.1

Replace the variable $x$ with $\frac{11 π}{6}$ in the expression.

$f (\frac{11 π}{6}) = \frac{\cos (\frac{11 π}{6})}{2 + \sin (\frac{11 π}{6})}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify the numerator.

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Step 4.2.1.1

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

$f (\frac{11 π}{6}) = \frac{\cos (\frac{π}{6})}{2 + \sin (\frac{11 π}{6})}$

Step 4.2.1.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$f (\frac{11 π}{6}) = \frac{\frac{\sqrt{3}}{2}}{2 + \sin (\frac{11 π}{6})}$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

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

$f (\frac{11 π}{6}) = \frac{\frac{\sqrt{3}}{2}}{2 - \sin (\frac{π}{6})}$

Step 4.2.2.2

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

$f (\frac{11 π}{6}) = \frac{\frac{\sqrt{3}}{2}}{2 - \frac{1}{2}}$

Step 4.2.2.3

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

$f (\frac{11 π}{6}) = \frac{\frac{\sqrt{3}}{2}}{2 \cdot \frac{2}{2} - \frac{1}{2}}$

Step 4.2.2.4

Combine $2$ and $\frac{2}{2}$ .

$f (\frac{11 π}{6}) = \frac{\frac{\sqrt{3}}{2}}{\frac{2 \cdot 2}{2} - \frac{1}{2}}$

Step 4.2.2.5

Combine the numerators over the common denominator.

$f (\frac{11 π}{6}) = \frac{\frac{\sqrt{3}}{2}}{\frac{2 \cdot 2 - 1}{2}}$

Step 4.2.2.6

Simplify the numerator.

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Step 4.2.2.6.1

Multiply $2$ by $2$ .

$f (\frac{11 π}{6}) = \frac{\frac{\sqrt{3}}{2}}{\frac{4 - 1}{2}}$

Step 4.2.2.6.2

Subtract $1$ from $4$ .

$f (\frac{11 π}{6}) = \frac{\frac{\sqrt{3}}{2}}{\frac{3}{2}}$

Step 4.2.3

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{11 π}{6}) = \frac{\sqrt{3}}{2} \cdot \frac{2}{3}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

$f (\frac{11 π}{6}) = \frac{\sqrt{3}}{2} \cdot \frac{2}{3}$

Step 4.2.4.2

Rewrite the expression.

$f (\frac{11 π}{6}) = \sqrt{3} (\frac{1}{3})$

Step 4.2.5

Combine $\sqrt{3}$ and $\frac{1}{3}$ .

$f (\frac{11 π}{6}) = \frac{\sqrt{3}}{3}$

Step 4.2.6

The final answer is $\frac{\sqrt{3}}{3}$ .

$\frac{\sqrt{3}}{3}$

Step 5

The horizontal tangent lines on function $f (x) = \frac{\cos (x)}{2 + \sin (x)}$ are $y = - \frac{\sqrt{3}}{3}, y = \frac{\sqrt{3}}{3}$ .

$y = - \frac{\sqrt{3}}{3}, y = \frac{\sqrt{3}}{3}$

Step 6