Calculus Examples

$2 \cos (2 x)$

Step 1

Find the derivative.

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Step 1.1

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

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

Step 1.2

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 $g (x) = 2 x$ .

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 1$ by $2$ .

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $2$ by $- 2$ .

$- 4 \sin (2 x) \frac{d}{d x} [x]$

Step 1.3.4

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

$- 4 \sin (2 x) \cdot 1$

Step 1.3.5

Multiply $- 4$ by $1$ .

$- 4 \sin (2 x)$

Step 2

Set the derivative equal to $0$ then solve the equation $- 4 \sin (2 x) = 0$ .

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Step 2.1

Divide each term in $- 4 \sin (2 x) = 0$ by $- 4$ and simplify.

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Step 2.1.1

Divide each term in $- 4 \sin (2 x) = 0$ by $- 4$ .

$\frac{- 4 \sin (2 x)}{- 4} = \frac{0}{- 4}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $- 4$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{- 4 \sin (2 x)}{- 4} = \frac{0}{- 4}$

Step 2.1.2.1.2

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

$\sin (2 x) = \frac{0}{- 4}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $- 4$ .

$\sin (2 x) = 0$

Step 2.2

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

$2 x = \arcsin (0)$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

$2 x = 0$

Step 2.4

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 2.4.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

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.

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $0$ by $2$ .

$x = 0$

Step 2.5

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

$2 x = π - 0$

Step 2.6

Solve for $x$ .

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Step 2.6.1

Simplify.

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Step 2.6.1.1

Multiply $- 1$ by $0$ .

$2 x = π + 0$

Step 2.6.1.2

Add $π$ and $0$ .

$2 x = π$

Step 2.6.2

Divide each term in $2 x = π$ by $2$ and simplify.

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Step 2.6.2.1

Divide each term in $2 x = π$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2}$

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.

$\frac{2 x}{2} = \frac{π}{2}$

Step 2.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{2}$

Step 2.7

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

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

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

Step 2.9

Consolidate the answers.

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

Step 3

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

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Step 3.1

$f (\frac{π}{2}) = 2 \cos (2 (\frac{π}{2}))$

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

Cancel the common factor.

$f (\frac{π}{2}) = 2 \cos (2 (\frac{π}{2}))$

Step 3.2.1.2

Rewrite the expression.

$f (\frac{π}{2}) = 2 \cos (π)$

Step 3.2.2

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 second quadrant.

$f (\frac{π}{2}) = 2 (- \cos (0))$

Step 3.2.3

The exact value of $\cos (0)$ is $1$ .

$f (\frac{π}{2}) = 2 (- 1 \cdot 1)$

Step 3.2.4

Multiply $2 (- 1 \cdot 1)$ .

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Step 3.2.4.1

Multiply $- 1$ by $1$ .

$f (\frac{π}{2}) = 2 \cdot - 1$

Step 3.2.4.2

Multiply $2$ by $- 1$ .

$f (\frac{π}{2}) = - 2$

Step 3.2.5

The final answer is $- 2$ .

$- 2$

Step 4

The horizontal tangent line on function $f (x) = 2 \cos (2 x)$ is $y = x = \frac{π n}{2}, y = - 2$ .

$y = x = \frac{π n}{2}, y = - 2$

Step 5