Calculus Examples

$\sin (2 x)$

Step 1

Find the derivative.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $2 x$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [2 x]$

Step 1.1.2

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

$\cos (u) \frac{d}{d x} [2 x]$

Step 1.1.3

Replace all occurrences of $u$ with $2 x$ .

$\cos (2 x) \frac{d}{d x} [2 x]$

Step 1.2

Differentiate.

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Step 1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$\cos (2 x) (2 \frac{d}{d x} [x])$

Step 1.2.2

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

$\cos (2 x) (2 \cdot 1)$

Step 1.2.3

Simplify the expression.

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Step 1.2.3.1

Multiply $2$ by $1$ .

$\cos (2 x) \cdot 2$

Step 1.2.3.2

Move $2$ to the left of $\cos (2 x)$ .

$2 \cos (2 x)$

Step 2

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

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Step 2.1

Divide each term in $2 \cos (2 x) = 0$ by $2$ and simplify.

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Step 2.1.1

Divide each term in $2 \cos (2 x) = 0$ by $2$ .

$\frac{2 \cos (2 x)}{2} = \frac{0}{2}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{2 \cos (2 x)}{2} = \frac{0}{2}$

Step 2.1.2.1.2

Divide $\cos (2 x)$ by $1$ .

$\cos (2 x) = \frac{0}{2}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $2$ .

$\cos (2 x) = 0$

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

Solve the original function $f (x) = \sin (2 x)$ at $x = \frac{π}{4}$ .

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Step 3.1

Replace the variable $x$ with $\frac{π}{4}$ in the expression.

$f (\frac{π}{4}) = \sin (2 (\frac{π}{4}))$

Step 3.2

Simplify the result.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Factor $2$ out of $4$ .

$f (\frac{π}{4}) = \sin (2 (\frac{π}{2 (2)}))$

Step 3.2.1.2

Cancel the common factor.

$f (\frac{π}{4}) = \sin (2 (\frac{π}{2 \cdot 2}))$

Step 3.2.1.3

Rewrite the expression.

$f (\frac{π}{4}) = \sin (\frac{π}{2})$

Step 3.2.2

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

$f (\frac{π}{4}) = 1$

Step 3.2.3

The final answer is $1$ .

$1$

Step 4

Solve the original function $f (x) = \sin (2 x)$ at $x = \frac{π}{4} + \frac{π}{2}$ .

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Step 4.1

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

$f (\frac{π}{4} + \frac{π}{2}) = \sin (2 (\frac{π}{4} + \frac{π}{2}))$

Step 4.2

Simplify the result.

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Step 4.2.1

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

$f (\frac{π}{4} + \frac{π}{2}) = \sin (2 (\frac{π}{4} + \frac{π}{2} \cdot \frac{2}{2}))$

Step 4.2.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.2.1

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

$f (\frac{π}{4} + \frac{π}{2}) = \sin (2 (\frac{π}{4} + \frac{π \cdot 2}{2 \cdot 2}))$

Step 4.2.2.2

Multiply $2$ by $2$ .

$f (\frac{π}{4} + \frac{π}{2}) = \sin (2 (\frac{π}{4} + \frac{π \cdot 2}{4}))$

Step 4.2.3

Combine the numerators over the common denominator.

$f (\frac{π}{4} + \frac{π}{2}) = \sin (2 (\frac{π + π \cdot 2}{4}))$

Step 4.2.4

Simplify the numerator.

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Step 4.2.4.1

Move $2$ to the left of $π$ .

$f (\frac{π}{4} + \frac{π}{2}) = \sin (2 (\frac{π + 2 \cdot π}{4}))$

Step 4.2.4.2

Add $π$ and $2 π$ .

$f (\frac{π}{4} + \frac{π}{2}) = \sin (2 (\frac{3 π}{4}))$

Step 4.2.5

Cancel the common factor of $2$ .

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Step 4.2.5.1

Factor $2$ out of $4$ .

$f (\frac{π}{4} + \frac{π}{2}) = \sin (2 (\frac{3 π}{2 (2)}))$

Step 4.2.5.2

Cancel the common factor.

$f (\frac{π}{4} + \frac{π}{2}) = \sin (2 (\frac{3 π}{2 \cdot 2}))$

Step 4.2.5.3

Rewrite the expression.

$f (\frac{π}{4} + \frac{π}{2}) = \sin (\frac{3 π}{2})$

Step 4.2.6

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.

$f (\frac{π}{4} + \frac{π}{2}) = - \sin (\frac{π}{2})$

Step 4.2.7

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

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

Step 4.2.8

Multiply $- 1$ by $1$ .

$f (\frac{π}{4} + \frac{π}{2}) = - 1$

Step 4.2.9

The final answer is $- 1$ .

$- 1$

Step 5

The horizontal tangent line on function $f (x) = \sin (2 x)$ is $y = 1, y = - 1$ .

$y = 1, y = - 1$

Step 6