Calculus Examples

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

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (x)]$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Multiply $- 1$ by $- 1$ .

$\cos (x) + 1 \sin (x)$

Step 1.1.3.4

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Divide each term in the equation by $\cos (x)$ .

$\frac{\cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 2.3

Cancel the common factor of $\cos (x)$ .

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Step 2.3.1

Cancel the common factor.

$\frac{\cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 2.3.2

Rewrite the expression.

$1 + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 2.4

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$1 + \tan (x) = \frac{0}{\cos (x)}$

Step 2.5

Separate fractions.

$1 + \tan (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 2.6

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$1 + \tan (x) = \frac{0}{1} \cdot \sec (x)$

Step 2.7

Divide $0$ by $1$ .

$1 + \tan (x) = 0 \sec (x)$

Step 2.8

Multiply $0$ by $\sec (x)$ .

$1 + \tan (x) = 0$

Step 2.9

Subtract $1$ from both sides of the equation.

$\tan (x) = - 1$

Step 2.10

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

$x = \arctan (- 1)$

Step 2.11

Simplify the right side.

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Step 2.11.1

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$x = - \frac{π}{4}$

Step 2.12

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the third quadrant.

$x = - \frac{π}{4} - π$

Step 2.13

Simplify the expression to find the second solution.

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Step 2.13.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 2.13.2

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{π}{4} - π$ .

$x = \frac{3 π}{4}$

Step 2.14

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

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Step 2.14.1

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

$\frac{π}{| b |}$

Step 2.14.2

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

$\frac{π}{| 1 |}$

Step 2.14.3

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

$\frac{π}{1}$

Step 2.14.4

Divide $π$ by $1$ .

$π$

Step 2.15

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

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Step 2.15.1

Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

Step 2.15.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 2.15.3

Combine fractions.

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Step 2.15.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 2.15.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 2.15.4

Simplify the numerator.

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Step 2.15.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 2.15.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 2.15.5

List the new angles.

$x = \frac{3 π}{4}$

Step 2.16

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

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

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $\sin (x) - \cos (x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{3 π}{4}$ .

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Step 4.1.1

Substitute $\frac{3 π}{4}$ for $x$ .

$\sin (\frac{3 π}{4}) - \cos (\frac{3 π}{4})$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$\sin (\frac{π}{4}) - \cos (\frac{3 π}{4})$

Step 4.1.2.1.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} - \cos (\frac{3 π}{4})$

Step 4.1.2.1.3

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.

$\frac{\sqrt{2}}{2} - - \cos (\frac{π}{4})$

Step 4.1.2.1.4

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} - - \frac{\sqrt{2}}{2}$

Step 4.1.2.1.5

Multiply $- - \frac{\sqrt{2}}{2}$ .

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Step 4.1.2.1.5.1

Multiply $- 1$ by $- 1$ .

$\frac{\sqrt{2}}{2} + 1 \frac{\sqrt{2}}{2}$

Step 4.1.2.1.5.2

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

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}$

Step 4.1.2.2

Simplify terms.

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Step 4.1.2.2.1

Combine the numerators over the common denominator.

$\frac{\sqrt{2} + \sqrt{2}}{2}$

Step 4.1.2.2.2

Add $\sqrt{2}$ and $\sqrt{2}$ .

$\frac{2 \sqrt{2}}{2}$

Step 4.1.2.2.3

Cancel the common factor of $2$ .

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Step 4.1.2.2.3.1

Cancel the common factor.

$\frac{2 \sqrt{2}}{2}$

Step 4.1.2.2.3.2

Divide $\sqrt{2}$ by $1$ .

$\sqrt{2}$

Step 4.2

Evaluate at $x = \frac{7 π}{4}$ .

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Step 4.2.1

Substitute $\frac{7 π}{4}$ for $x$ .

$\sin (\frac{7 π}{4}) - \cos (\frac{7 π}{4})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

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.

$- \sin (\frac{π}{4}) - \cos (\frac{7 π}{4})$

Step 4.2.2.1.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{2}}{2} - \cos (\frac{7 π}{4})$

Step 4.2.2.1.3

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

$- \frac{\sqrt{2}}{2} - \cos (\frac{π}{4})$

Step 4.2.2.1.4

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}$

Step 4.2.2.2

Simplify terms.

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Step 4.2.2.2.1

Combine the numerators over the common denominator.

$\frac{- \sqrt{2} - \sqrt{2}}{2}$

Step 4.2.2.2.2

Subtract $\sqrt{2}$ from $- \sqrt{2}$ .

$\frac{- 2 \sqrt{2}}{2}$

Step 4.2.2.2.3

Cancel the common factor of $- 2$ and $2$ .

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Step 4.2.2.2.3.1

Factor $2$ out of $- 2 \sqrt{2}$ .

$\frac{2 (- \sqrt{2})}{2}$

Step 4.2.2.2.3.2

Cancel the common factors.

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Step 4.2.2.2.3.2.1

Factor $2$ out of $2$ .

$\frac{2 (- \sqrt{2})}{2 (1)}$

Step 4.2.2.2.3.2.2

Cancel the common factor.

$\frac{2 (- \sqrt{2})}{2 \cdot 1}$

Step 4.2.2.2.3.2.3

Rewrite the expression.

$\frac{- \sqrt{2}}{1}$

Step 4.2.2.2.3.2.4

Divide $- \sqrt{2}$ by $1$ .

$- \sqrt{2}$

Step 4.3

List all of the points.

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

Step 5