Calculus Examples

$f (x) = \sec (\frac{π x}{4})$

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

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = \frac{π x}{4}$ .

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Step 1.1.1.1

To apply the Chain Rule, set $u$ as $\frac{π x}{4}$ .

$\frac{d}{d u} [\sec (u)] \frac{d}{d x} [\frac{π x}{4}]$

Step 1.1.1.2

The derivative of $\sec (u)$ with respect to $u$ is $\sec (u) \tan (u)$ .

$\sec (u) \tan (u) \frac{d}{d x} [\frac{π x}{4}]$

Step 1.1.1.3

Replace all occurrences of $u$ with $\frac{π x}{4}$ .

$\sec (\frac{π x}{4}) \tan (\frac{π x}{4}) \frac{d}{d x} [\frac{π x}{4}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

Since $\frac{π}{4}$ is constant with respect to $x$ , the derivative of $\frac{π x}{4}$ with respect to $x$ is $\frac{π}{4} \frac{d}{d x} [x]$ .

$\sec (\frac{π x}{4}) \tan (\frac{π x}{4}) (\frac{π}{4} \frac{d}{d x} [x])$

Step 1.1.2.2

Combine fractions.

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Step 1.1.2.2.1

Combine $\frac{π}{4}$ and $\sec (\frac{π x}{4})$ .

$\frac{π \sec (\frac{π x}{4})}{4} \tan (\frac{π x}{4}) \frac{d}{d x} [x]$

Step 1.1.2.2.2

Combine $\frac{π \sec (\frac{π x}{4})}{4}$ and $\tan (\frac{π x}{4})$ .

$\frac{π \sec (\frac{π x}{4}) \tan (\frac{π x}{4})}{4} \frac{d}{d x} [x]$

Step 1.1.2.3

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

$\frac{π \sec (\frac{π x}{4}) \tan (\frac{π x}{4})}{4} \cdot 1$

Step 1.1.2.4

Multiply $\frac{π \sec (\frac{π x}{4}) \tan (\frac{π x}{4})}{4}$ by $1$ .

$f' (x) = \frac{π \sec (\frac{π x}{4}) \tan (\frac{π x}{4})}{4}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{π \sec (\frac{π x}{4}) \tan (\frac{π x}{4})}{4}$ .

$\frac{π \sec (\frac{π x}{4}) \tan (\frac{π x}{4})}{4}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{π \sec (\frac{π x}{4}) \tan (\frac{π x}{4})}{4} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{π \sec (\frac{π x}{4}) \tan (\frac{π x}{4})}{4} = 0$

Step 2.2

Set the numerator equal to zero.

$π \sec (\frac{π x}{4}) \tan (\frac{π x}{4}) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\sec (\frac{π x}{4}) = 0$

$\tan (\frac{π x}{4}) = 0$

Step 2.3.2

Set $\sec (\frac{π x}{4})$ equal to $0$ and solve for $x$ .

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Step 2.3.2.1

Set $\sec (\frac{π x}{4})$ equal to $0$ .

$\sec (\frac{π x}{4}) = 0$

Step 2.3.2.2

The range of secant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

Step 2.3.3

Set $\tan (\frac{π x}{4})$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $\tan (\frac{π x}{4})$ equal to $0$ .

$\tan (\frac{π x}{4}) = 0$

Step 2.3.3.2

Solve $\tan (\frac{π x}{4}) = 0$ for $x$ .

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Step 2.3.3.2.1

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

$\frac{π x}{4} = \arctan (0)$

Step 2.3.3.2.2

Simplify the right side.

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Step 2.3.3.2.2.1

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

$\frac{π x}{4} = 0$

Step 2.3.3.2.3

Set the numerator equal to zero.

$π x = 0$

Step 2.3.3.2.4

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

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Step 2.3.3.2.4.1

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

$\frac{π x}{π} = \frac{0}{π}$

Step 2.3.3.2.4.2

Simplify the left side.

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Step 2.3.3.2.4.2.1

Cancel the common factor of $π$ .

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Step 2.3.3.2.4.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{0}{π}$

Step 2.3.3.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{π}$

Step 2.3.3.2.4.3

Simplify the right side.

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Step 2.3.3.2.4.3.1

Divide $0$ by $π$ .

$x = 0$

Step 2.3.3.2.5

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

$\frac{π x}{4} = π + 0$

Step 2.3.3.2.6

Solve for $x$ .

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Step 2.3.3.2.6.1

Multiply both sides of the equation by $\frac{4}{π}$ .

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} (π + 0)$

Step 2.3.3.2.6.2

Simplify both sides of the equation.

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Step 2.3.3.2.6.2.1

Simplify the left side.

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Step 2.3.3.2.6.2.1.1

Simplify $\frac{4}{π} \cdot \frac{π x}{4}$ .

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Step 2.3.3.2.6.2.1.1.1

Cancel the common factor of $4$ .

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Step 2.3.3.2.6.2.1.1.1.1

Cancel the common factor.

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} (π + 0)$

Step 2.3.3.2.6.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{4}{π} (π + 0)$

Step 2.3.3.2.6.2.1.1.2

Cancel the common factor of $π$ .

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Step 2.3.3.2.6.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{4}{π} (π + 0)$

Step 2.3.3.2.6.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{4}{π} (π + 0)$

Step 2.3.3.2.6.2.1.1.2.3

Rewrite the expression.

$x = \frac{4}{π} (π + 0)$

Step 2.3.3.2.6.2.2

Simplify the right side.

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Step 2.3.3.2.6.2.2.1

Simplify $\frac{4}{π} (π + 0)$ .

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Step 2.3.3.2.6.2.2.1.1

Add $π$ and $0$ .

$x = \frac{4}{π} π$

Step 2.3.3.2.6.2.2.1.2

Cancel the common factor of $π$ .

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Step 2.3.3.2.6.2.2.1.2.1

Cancel the common factor.

$x = \frac{4}{π} π$

Step 2.3.3.2.6.2.2.1.2.2

Rewrite the expression.

$x = 4$

Step 2.3.3.2.7

Find the period of $\tan (\frac{π x}{4})$ .

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Step 2.3.3.2.7.1

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

$\frac{π}{| b |}$

Step 2.3.3.2.7.2

Replace $b$ with $\frac{π}{4}$ in the formula for period.

$\frac{π}{| \frac{π}{4} |}$

Step 2.3.3.2.7.3

$\frac{π}{4}$ is approximately $0.78539816$ which is positive so remove the absolute value

$\frac{π}{\frac{π}{4}}$

Step 2.3.3.2.7.4

Multiply the numerator by the reciprocal of the denominator.

$π \frac{4}{π}$

Step 2.3.3.2.7.5

Cancel the common factor of $π$ .

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Step 2.3.3.2.7.5.1

Cancel the common factor.

$π \frac{4}{π}$

Step 2.3.3.2.7.5.2

Rewrite the expression.

$4$

Step 2.3.3.2.8

The period of the $\tan (\frac{π x}{4})$ function is $4$ so values will repeat every $4$ radians in both directions.

$x = 4 n, 4 + 4 n$ , for any integer $n$

Step 2.3.4

The final solution is all the values that make $π \sec (\frac{π x}{4}) \tan (\frac{π x}{4}) = 0$ true.

$x = 4 n, 4 + 4 n$ , for any integer $n$

Step 2.4

Consolidate the answers.

$x = 4 n$ , 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 $\sec (\frac{π x}{4})$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

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

Step 3.2

Solve for $x$ .

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Step 3.2.1

Multiply both sides of the equation by $\frac{4}{π}$ .

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} (\frac{π}{2} + π n)$

Step 3.2.2

Simplify both sides of the equation.

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Step 3.2.2.1

Simplify the left side.

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Step 3.2.2.1.1

Simplify $\frac{4}{π} \cdot \frac{π x}{4}$ .

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Step 3.2.2.1.1.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1.1.1

Cancel the common factor.

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} (\frac{π}{2} + π n)$

Step 3.2.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{4}{π} (\frac{π}{2} + π n)$

Step 3.2.2.1.1.2

Cancel the common factor of $π$ .

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Step 3.2.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{4}{π} (\frac{π}{2} + π n)$

Step 3.2.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{4}{π} (\frac{π}{2} + π n)$

Step 3.2.2.1.1.2.3

Rewrite the expression.

$x = \frac{4}{π} (\frac{π}{2} + π n)$

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Simplify $\frac{4}{π} (\frac{π}{2} + π n)$ .

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Step 3.2.2.2.1.1

Apply the distributive property.

$x = \frac{4}{π} \cdot \frac{π}{2} + \frac{4}{π} (π n)$

Step 3.2.2.2.1.2

Cancel the common factor of $2$ .

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Step 3.2.2.2.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 (2)}{π} \cdot \frac{π}{2} + \frac{4}{π} (π n)$

Step 3.2.2.2.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 2}{π} \cdot \frac{π}{2} + \frac{4}{π} (π n)$

Step 3.2.2.2.1.2.3

Rewrite the expression.

$x = \frac{2}{π} π + \frac{4}{π} (π n)$

Step 3.2.2.2.1.3

Cancel the common factor of $π$ .

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Step 3.2.2.2.1.3.1

Cancel the common factor.

$x = \frac{2}{π} π + \frac{4}{π} (π n)$

Step 3.2.2.2.1.3.2

Rewrite the expression.

$x = 2 + \frac{4}{π} (π n)$

Step 3.2.2.2.1.4

Cancel the common factor of $π$ .

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Step 3.2.2.2.1.4.1

Factor $π$ out of $π n$ .

$x = 2 + \frac{4}{π} (π (n))$

Step 3.2.2.2.1.4.2

Cancel the common factor.

$x = 2 + \frac{4}{π} (π n)$

Step 3.2.2.2.1.4.3

Rewrite the expression.

$x = 2 + 4 n$

Step 3.2.3

Reorder $2$ and $4 n$ .

$x = 4 n + 2$

Step 3.3

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 $0$ .

${x | x = 4 n + 2}$ $n$ , for any integer $n$

Step 4

Evaluate $\sec (\frac{π x}{4})$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$\sec (\frac{π (0)}{4})$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Cancel the common factor of $0$ and $4$ .

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Step 4.1.2.1.1

Factor $4$ out of $π (0)$ .

$\sec (\frac{4 (π \cdot (0))}{4})$

Step 4.1.2.1.2

Cancel the common factors.

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Step 4.1.2.1.2.1

Factor $4$ out of $4$ .

$\sec (\frac{4 (π \cdot (0))}{4 (1)})$

Step 4.1.2.1.2.2

Cancel the common factor.

$\sec (\frac{4 (π \cdot (0))}{4 \cdot 1})$

Step 4.1.2.1.2.3

Rewrite the expression.

$\sec (\frac{π \cdot (0)}{1})$

Step 4.1.2.1.2.4

Divide $π \cdot (0)$ by $1$ .

$\sec (π \cdot (0))$

Step 4.1.2.2

Multiply $π$ by $0$ .

$\sec (0)$

Step 4.1.2.3

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

$1$

Step 4.2

Evaluate at $x = 4$ .

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Step 4.2.1

Substitute $4$ for $x$ .

$\sec (\frac{π (4)}{4})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\sec (\frac{π \cdot 4}{4})$

Step 4.2.2.1.2

Divide $π$ by $1$ .

$\sec (π)$

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 secant is negative in the second quadrant.

$- \sec (0)$

Step 4.2.2.3

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

$- 1 \cdot 1$

Step 4.2.2.4

Multiply $- 1$ by $1$ .

$- 1$

Step 4.3

List all of the points.

$(0 + 8 n, 1), (4 + 8 n, - 1)$ , for any integer $n$

Step 5