Calculus Examples

$f (x) = \sin (x) \cos (x) + 9$

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 $\sin (x) \cos (x) + 9$ with respect to $x$ is $\frac{d}{d x} [\sin (x) \cos (x)] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [\sin (x) \cos (x)] + \frac{d}{d x} [9]$

Step 1.1.2

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

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Step 1.1.2.1

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

$\sin (x) \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [9]$

Step 1.1.2.2

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

$\sin (x) (- \sin (x)) + \cos (x) \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [9]$

Step 1.1.2.3

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

$\sin (x) (- \sin (x)) + \cos (x) \cos (x) + \frac{d}{d x} [9]$

Step 1.1.2.4

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

$- (\sin^{1} (x) \sin (x)) + \cos (x) \cos (x) + \frac{d}{d x} [9]$

Step 1.1.2.5

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

$- (\sin^{1} (x) \sin^{1} (x)) + \cos (x) \cos (x) + \frac{d}{d x} [9]$

Step 1.1.2.6

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

$- {\sin (x)}^{1 + 1} + \cos (x) \cos (x) + \frac{d}{d x} [9]$

Step 1.1.2.7

Add $1$ and $1$ .

$- \sin^{2} (x) + \cos (x) \cos (x) + \frac{d}{d x} [9]$

Step 1.1.2.8

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

$- \sin^{2} (x) + \cos^{1} (x) \cos (x) + \frac{d}{d x} [9]$

Step 1.1.2.9

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

$- \sin^{2} (x) + \cos^{1} (x) \cos^{1} (x) + \frac{d}{d x} [9]$

Step 1.1.2.10

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

$- \sin^{2} (x) + {\cos (x)}^{1 + 1} + \frac{d}{d x} [9]$

Step 1.1.2.11

Add $1$ and $1$ .

$- \sin^{2} (x) + \cos^{2} (x) + \frac{d}{d x} [9]$

Step 1.1.3

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$- \sin^{2} (x) + \cos^{2} (x) + 0$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Add $- \sin^{2} (x) + \cos^{2} (x)$ and $0$ .

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

Step 1.1.4.2

Reorder $- \sin^{2} (x)$ and $\cos^{2} (x)$ .

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

Step 1.1.4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (x)$ and $b = \sin (x)$ .

$(\cos (x) + \sin (x)) (\cos (x) - \sin (x))$

Step 1.1.4.4

Expand $(\cos (x) + \sin (x)) (\cos (x) - \sin (x))$ using the FOIL Method.

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Step 1.1.4.4.1

Apply the distributive property.

$\cos (x) (\cos (x) - \sin (x)) + \sin (x) (\cos (x) - \sin (x))$

Step 1.1.4.4.2

Apply the distributive property.

$\cos (x) \cos (x) + \cos (x) (- \sin (x)) + \sin (x) (\cos (x) - \sin (x))$

Step 1.1.4.4.3

Apply the distributive property.

$\cos (x) \cos (x) + \cos (x) (- \sin (x)) + \sin (x) \cos (x) + \sin (x) (- \sin (x))$

Step 1.1.4.5

Combine the opposite terms in $\cos (x) \cos (x) + \cos (x) (- \sin (x)) + \sin (x) \cos (x) + \sin (x) (- \sin (x))$ .

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Step 1.1.4.5.1

Reorder the factors in the terms $\cos (x) (- \sin (x))$ and $\sin (x) \cos (x)$ .

$\cos (x) \cos (x) - \cos (x) \sin (x) + \cos (x) \sin (x) + \sin (x) (- \sin (x))$

Step 1.1.4.5.2

Add $- \cos (x) \sin (x)$ and $\cos (x) \sin (x)$ .

$\cos (x) \cos (x) + 0 + \sin (x) (- \sin (x))$

Step 1.1.4.5.3

Add $\cos (x) \cos (x)$ and $0$ .

$\cos (x) \cos (x) + \sin (x) (- \sin (x))$

Step 1.1.4.6

Simplify each term.

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Step 1.1.4.6.1

Multiply $\cos (x) \cos (x)$ .

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Step 1.1.4.6.1.1

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

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

Step 1.1.4.6.1.2

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

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

Step 1.1.4.6.1.3

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

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

Step 1.1.4.6.1.4

Add $1$ and $1$ .

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

Step 1.1.4.6.2

Rewrite using the commutative property of multiplication.

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

Step 1.1.4.6.3

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

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Step 1.1.4.6.3.1

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

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

Step 1.1.4.6.3.2

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

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

Step 1.1.4.6.3.3

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

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

Step 1.1.4.6.3.4

Add $1$ and $1$ .

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

Step 1.1.4.7

Apply the cosine double-angle identity.

$f' (x) = \cos (2 x)$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\cos (2 x)$ .

$\cos (2 x)$

Step 2

Set the first derivative equal to $0$ then solve the equation $\cos (2 x) = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\cos (2 x) = 0$

Step 2.2

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

$2 x = \arccos (0)$

Step 2.3

Simplify the right side.

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Step 2.3.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$2 x = \frac{π}{2}$

Step 2.4

Divide each term in $2 x = \frac{π}{2}$ by $2$ and simplify.

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Step 2.4.1

Divide each term in $2 x = \frac{π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 2.4.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Step 2.4.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

$x = \frac{π}{2 \cdot 2}$

Step 2.4.3.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Step 2.5

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

$2 x = 2 π - \frac{π}{2}$

Step 2.6

Solve for $x$ .

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Step 2.6.1

Simplify.

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Step 2.6.1.1

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

$2 x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 2.6.1.2

Combine $2 π$ and $\frac{2}{2}$ .

$2 x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 2.6.1.3

Combine the numerators over the common denominator.

$2 x = \frac{2 π \cdot 2 - π}{2}$

Step 2.6.1.4

Multiply $2$ by $2$ .

$2 x = \frac{4 π - π}{2}$

Step 2.6.1.5

Subtract $π$ from $4 π$ .

$2 x = \frac{3 π}{2}$

Step 2.6.2

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ and simplify.

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Step 2.6.2.1

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2}$

Step 2.6.2.2

Simplify the left side.

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Step 2.6.2.2.1

Cancel the common factor of $2$ .

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Step 2.6.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2}$

Step 2.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{2}$

Step 2.6.2.3

Simplify the right side.

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Step 2.6.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{2}$

Step 2.6.2.3.2

Multiply $\frac{3 π}{2} \cdot \frac{1}{2}$ .

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Step 2.6.2.3.2.1

Multiply $\frac{3 π}{2}$ by $\frac{1}{2}$ .

$x = \frac{3 π}{2 \cdot 2}$

Step 2.6.2.3.2.2

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

Step 2.7

Find the period of $\cos (2 x)$ .

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Step 2.7.1

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

$\frac{2 π}{| b |}$

Step 2.7.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 2.7.4

Cancel the common factor of $2$ .

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Step 2.7.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 2.7.4.2

Divide $π$ by $1$ .

$π$

Step 2.8

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

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

Step 2.9

Consolidate the answers.

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

Step 3

The values which make the derivative equal to $0$ are $\frac{π}{4} + \frac{π n}{2}$ .

$\frac{π}{4} + \frac{π n}{2}$

Step 4

After finding the point that makes the derivative $f' (x) = \cos (2 x)$ equal to $0$ or undefined, the interval to check where $f (x) = \sin (x) \cos (x) + 9$ is increasing and where it is decreasing is $(- \infty, \frac{π}{4} + \frac{π n}{2}) \cup (\frac{π}{4} + \frac{π n}{2}, \infty)$ .

$(- \infty, \frac{π}{4} + \frac{π n}{2}) \cup (\frac{π}{4} + \frac{π n}{2}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{π}{4} + \frac{π n}{2})$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $\frac{π}{4} + \frac{π n}{2} - 1$ in the expression.

$f' (\frac{π}{4} + \frac{π n}{2} - 1) = \cos (2 (\frac{π}{4} + \frac{π n}{2} - 1))$

Step 5.2

Simplify the result.

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Step 5.2.1

Apply the distributive property.

$f' (\frac{π}{4} + \frac{π n}{2} - 1) = \cos (2 (\frac{π}{4}) + 2 (\frac{π n}{2}) + 2 \cdot - 1)$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Factor $2$ out of $4$ .

$f' (\frac{π}{4} + \frac{π n}{2} - 1) = \cos (2 (\frac{π}{2 (2)}) + 2 (\frac{π n}{2}) + 2 \cdot - 1)$

Step 5.2.2.1.2

Cancel the common factor.

$f' (\frac{π}{4} + \frac{π n}{2} - 1) = \cos (2 (\frac{π}{2 \cdot 2}) + 2 (\frac{π n}{2}) + 2 \cdot - 1)$

Step 5.2.2.1.3

Rewrite the expression.

$f' (\frac{π}{4} + \frac{π n}{2} - 1) = \cos (\frac{π}{2} + 2 (\frac{π n}{2}) + 2 \cdot - 1)$

Step 5.2.2.2

Cancel the common factor of $2$ .

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Step 5.2.2.2.1

Cancel the common factor.

$f' (\frac{π}{4} + \frac{π n}{2} - 1) = \cos (\frac{π}{2} + 2 (\frac{π n}{2}) + 2 \cdot - 1)$

Step 5.2.2.2.2

Rewrite the expression.

$f' (\frac{π}{4} + \frac{π n}{2} - 1) = \cos (\frac{π}{2} + π n + 2 \cdot - 1)$

Step 5.2.2.3

Multiply $2$ by $- 1$ .

$f' (\frac{π}{4} + \frac{π n}{2} - 1) = \cos (\frac{π}{2} + π n - 2)$

Step 5.2.3

The final answer is $\cos (\frac{π}{2} + π n - 2)$ .

$\cos (\frac{π}{2} + π n - 2)$

Step 5.3

At $x = \frac{π}{4} + \frac{π n}{2} - 1$ the derivative is $\cos (\frac{π}{2} + π n - 2)$ . Since this is negative, the function is decreasing on $(- \infty, \frac{π}{4} + \frac{π n}{2})$ .

Decreasing on $(- \infty, \frac{π}{4} + \frac{π n}{2})$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(\frac{π}{4} + \frac{π n}{2}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $\frac{π}{4} + \frac{π n}{2} + 1$ in the expression.

$f' (\frac{π}{4} + \frac{π n}{2} + 1) = \cos (2 (\frac{π}{4} + \frac{π n}{2} + 1))$

Step 6.2

Simplify the result.

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Step 6.2.1

Apply the distributive property.

$f' (\frac{π}{4} + \frac{π n}{2} + 1) = \cos (2 (\frac{π}{4}) + 2 (\frac{π n}{2}) + 2 \cdot 1)$

Step 6.2.2

Simplify.

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Step 6.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1

Factor $2$ out of $4$ .

$f' (\frac{π}{4} + \frac{π n}{2} + 1) = \cos (2 (\frac{π}{2 (2)}) + 2 (\frac{π n}{2}) + 2 \cdot 1)$

Step 6.2.2.1.2

Cancel the common factor.

$f' (\frac{π}{4} + \frac{π n}{2} + 1) = \cos (2 (\frac{π}{2 \cdot 2}) + 2 (\frac{π n}{2}) + 2 \cdot 1)$

Step 6.2.2.1.3

Rewrite the expression.

$f' (\frac{π}{4} + \frac{π n}{2} + 1) = \cos (\frac{π}{2} + 2 (\frac{π n}{2}) + 2 \cdot 1)$

Step 6.2.2.2

Cancel the common factor of $2$ .

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Step 6.2.2.2.1

Cancel the common factor.

$f' (\frac{π}{4} + \frac{π n}{2} + 1) = \cos (\frac{π}{2} + 2 (\frac{π n}{2}) + 2 \cdot 1)$

Step 6.2.2.2.2

Rewrite the expression.

$f' (\frac{π}{4} + \frac{π n}{2} + 1) = \cos (\frac{π}{2} + π n + 2 \cdot 1)$

Step 6.2.2.3

Multiply $2$ by $1$ .

$f' (\frac{π}{4} + \frac{π n}{2} + 1) = \cos (\frac{π}{2} + π n + 2)$

Step 6.2.3

The final answer is $\cos (\frac{π}{2} + π n + 2)$ .

$\cos (\frac{π}{2} + π n + 2)$

Step 6.3

At $x = \frac{π}{4} + \frac{π n}{2} + 1$ the derivative is $\cos (\frac{π}{2} + π n + 2)$ . Since this is positive, the function is increasing on $(\frac{π}{4} + \frac{π n}{2}, \infty)$ .

Increasing on $(\frac{π}{4} + \frac{π n}{2}, \infty)$ since $f' (x) > 0$

Step 7

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

Increasing on: $(\frac{π}{4} + \frac{π n}{2}, \infty)$

Decreasing on: $(- \infty, \frac{π}{4} + \frac{π n}{2})$

Step 8