Calculus Examples

$y = 2 x - \tan (x)$

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 $2 x - \tan (x)$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \tan (x)]$ .

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

Step 1.1.2

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

$2 \cdot 1 + \frac{d}{d x} [- \tan (x)]$

Step 1.1.2.3

Multiply $2$ by $1$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$f' (x) = 2 - \sec^{2} (x)$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $2 - \sec^{2} (x)$ .

$2 - \sec^{2} (x)$

Step 2

Set the first derivative equal to $0$ then solve the equation $2 - \sec^{2} (x) = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$2 - \sec^{2} (x) = 0$

Step 2.2

Subtract $2$ from both sides of the equation.

$- \sec^{2} (x) = - 2$

Step 2.3

Divide each term in $- \sec^{2} (x) = - 2$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- \sec^{2} (x) = - 2$ by $- 1$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

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

Step 2.3.2.2

Divide $\sec^{2} (x)$ by $1$ .

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 2$ by $- 1$ .

$\sec^{2} (x) = 2$

Step 2.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sec (x) = \pm \sqrt{2}$

Step 2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$\sec (x) = \sqrt{2}$

Step 2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$\sec (x) = - \sqrt{2}$

Step 2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$\sec (x) = \sqrt{2}, - \sqrt{2}$

Step 2.6

Set up each of the solutions to solve for $x$ .

$\sec (x) = \sqrt{2}$

$\sec (x) = - \sqrt{2}$

Step 2.7

Solve for $x$ in $\sec (x) = \sqrt{2}$ .

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Step 2.7.1

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

$x = arcsec (\sqrt{2})$

Step 2.7.2

Simplify the right side.

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Step 2.7.2.1

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

$x = \frac{π}{4}$

Step 2.7.3

The secant 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.

$x = 2 π - \frac{π}{4}$

Step 2.7.4

Simplify $2 π - \frac{π}{4}$ .

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Step 2.7.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 2.7.4.2

Combine fractions.

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Step 2.7.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 2.7.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 2.7.4.3

Simplify the numerator.

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Step 2.7.4.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - π}{4}$

Step 2.7.4.3.2

Subtract $π$ from $8 π$ .

$x = \frac{7 π}{4}$

Step 2.7.5

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

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Step 2.7.5.1

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

$\frac{2 π}{| b |}$

Step 2.7.5.2

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

$\frac{2 π}{| 1 |}$

Step 2.7.5.3

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

$\frac{2 π}{1}$

Step 2.7.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.7.6

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

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 2.8

Solve for $x$ in $\sec (x) = - \sqrt{2}$ .

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Step 2.8.1

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

$x = arcsec (- \sqrt{2})$

Step 2.8.2

Simplify the right side.

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Step 2.8.2.1

The exact value of $arcsec (- \sqrt{2})$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 2.8.3

The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - \frac{3 π}{4}$

Step 2.8.4

Simplify $2 π - \frac{3 π}{4}$ .

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Step 2.8.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{3 π}{4}$

Step 2.8.4.2

Combine fractions.

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Step 2.8.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{3 π}{4}$

Step 2.8.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - 3 π}{4}$

Step 2.8.4.3

Simplify the numerator.

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Step 2.8.4.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - 3 π}{4}$

Step 2.8.4.3.2

Subtract $3 π$ from $8 π$ .

$x = \frac{5 π}{4}$

Step 2.8.5

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

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Step 2.8.5.1

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

$\frac{2 π}{| b |}$

Step 2.8.5.2

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

$\frac{2 π}{| 1 |}$

Step 2.8.5.3

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

$\frac{2 π}{1}$

Step 2.8.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.8.6

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

$x = \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$

Step 2.9

List all of the solutions.

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n, \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$

Step 2.10

Consolidate the answers.

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

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the argument in $\sec (x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

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

Step 3.2

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

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

Step 4

Evaluate $2 x - \tan (x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{π}{4}$ .

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Step 4.1.1

Substitute $\frac{π}{4}$ for $x$ .

$2 (\frac{π}{4}) - \tan (\frac{π}{4})$

Step 4.1.2

Simplify each term.

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Step 4.1.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.1.1

Factor $2$ out of $4$ .

$2 \frac{π}{2 (2)} - \tan (\frac{π}{4})$

Step 4.1.2.1.2

Cancel the common factor.

$2 \frac{π}{2 \cdot 2} - \tan (\frac{π}{4})$

Step 4.1.2.1.3

Rewrite the expression.

$\frac{π}{2} - \tan (\frac{π}{4})$

Step 4.1.2.2

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

$\frac{π}{2} - 1 \cdot 1$

Step 4.1.2.3

Multiply $- 1$ by $1$ .

$\frac{π}{2} - 1$

Step 4.2

Evaluate at $x = \frac{3 π}{4}$ .

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Step 4.2.1

Substitute $\frac{3 π}{4}$ for $x$ .

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

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Factor $2$ out of $4$ .

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

Step 4.2.2.1.2

Cancel the common factor.

$2 \frac{3 π}{2 \cdot 2} - \tan (\frac{3 π}{4})$

Step 4.2.2.1.3

Rewrite the expression.

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

Step 4.2.2.2

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

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

Step 4.2.2.3

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

$\frac{3 π}{2} - (- 1 \cdot 1)$

Step 4.2.2.4

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

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Step 4.2.2.4.1

Multiply $- 1$ by $1$ .

$\frac{3 π}{2} - - 1$

Step 4.2.2.4.2

Multiply $- 1$ by $- 1$ .

$\frac{3 π}{2} + 1$

Step 4.3

Evaluate at $x = \frac{5 π}{4}$ .

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Step 4.3.1

Substitute $\frac{5 π}{4}$ for $x$ .

$2 (\frac{5 π}{4}) - \tan (\frac{5 π}{4})$

Step 4.3.2

Simplify each term.

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Step 4.3.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1

Factor $2$ out of $4$ .

$2 \frac{5 π}{2 (2)} - \tan (\frac{5 π}{4})$

Step 4.3.2.1.2

Cancel the common factor.

$2 \frac{5 π}{2 \cdot 2} - \tan (\frac{5 π}{4})$

Step 4.3.2.1.3

Rewrite the expression.

$\frac{5 π}{2} - \tan (\frac{5 π}{4})$

Step 4.3.2.2

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

$\frac{5 π}{2} - \tan (\frac{π}{4})$

Step 4.3.2.3

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

$\frac{5 π}{2} - 1 \cdot 1$

Step 4.3.2.4

Multiply $- 1$ by $1$ .

$\frac{5 π}{2} - 1$

Step 4.4

Evaluate at $x = \frac{7 π}{4}$ .

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Step 4.4.1

Substitute $\frac{7 π}{4}$ for $x$ .

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

Step 4.4.2

Simplify each term.

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Step 4.4.2.1

Cancel the common factor of $2$ .

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Step 4.4.2.1.1

Factor $2$ out of $4$ .

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

Step 4.4.2.1.2

Cancel the common factor.

$2 \frac{7 π}{2 \cdot 2} - \tan (\frac{7 π}{4})$

Step 4.4.2.1.3

Rewrite the expression.

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

Step 4.4.2.2

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

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

Step 4.4.2.3

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

$\frac{7 π}{2} - (- 1 \cdot 1)$

Step 4.4.2.4

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

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Step 4.4.2.4.1

Multiply $- 1$ by $1$ .

$\frac{7 π}{2} - - 1$

Step 4.4.2.4.2

Multiply $- 1$ by $- 1$ .

$\frac{7 π}{2} + 1$

Step 4.5

Evaluate at $x = \frac{9 π}{4}$ .

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Step 4.5.1

Substitute $\frac{9 π}{4}$ for $x$ .

$2 (\frac{9 π}{4}) - \tan (\frac{9 π}{4})$

Step 4.5.2

Simplify each term.

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Step 4.5.2.1

Cancel the common factor of $2$ .

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Step 4.5.2.1.1

Factor $2$ out of $4$ .

$2 \frac{9 π}{2 (2)} - \tan (\frac{9 π}{4})$

Step 4.5.2.1.2

Cancel the common factor.

$2 \frac{9 π}{2 \cdot 2} - \tan (\frac{9 π}{4})$

Step 4.5.2.1.3

Rewrite the expression.

$\frac{9 π}{2} - \tan (\frac{9 π}{4})$

Step 4.5.2.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{9 π}{2} - \tan (\frac{π}{4})$

Step 4.5.2.3

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

$\frac{9 π}{2} - 1 \cdot 1$

Step 4.5.2.4

Multiply $- 1$ by $1$ .

$\frac{9 π}{2} - 1$

Step 4.6

List all of the points.

$(\frac{π}{4}, \frac{π}{2} - 1), (\frac{3 π}{4}, \frac{3 π}{2} + 1), (\frac{5 π}{4}, \frac{5 π}{2} - 1), (\frac{7 π}{4}, \frac{7 π}{2} + 1), (\frac{9 π}{4}, \frac{9 π}{2} - 1)$

Step 5