Calculus Examples

$2 \sec (θ) + \tan (θ)$

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

$f' (θ) = \frac{d}{d θ} (2 \sec (θ)) + \frac{d}{d θ} (\tan (θ))$

Step 1.1.2

Evaluate $\frac{d}{d θ} [2 \sec (θ)]$ .

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Step 1.1.2.1

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

$f' (θ) = 2 \frac{d}{d θ} (\sec (θ)) + \frac{d}{d θ} (\tan (θ))$

Step 1.1.2.2

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

$f' (θ) = 2 \sec (θ) \tan (θ) + \frac{d}{d θ} (\tan (θ))$

Step 1.1.3

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

$f' (θ) = 2 \sec (θ) \tan (θ) + \sec^{2} (θ)$

Step 1.2

The first derivative of $f (θ)$ with respect to $θ$ is $2 \sec (θ) \tan (θ) + \sec^{2} (θ)$ .

$2 \sec (θ) \tan (θ) + \sec^{2} (θ)$

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

$2 \sec (θ) \tan (θ) + \sec^{2} (θ) = 0$

Step 2.2

Replace the $\sec^{2} (θ)$ with $1 + \tan^{2} (θ)$ based on the $\tan^{2} (x) + 1 = \sec^{2} (x)$ identity.

$2 \sec (θ) \tan (θ) (1 + \tan^{2} (θ)) = 0$

Step 2.3

Apply the distributive property.

$2 \sec (θ) \tan (θ) \cdot 1 + 2 \sec (θ) \tan (θ) \tan^{2} (θ) = 0$

Step 2.4

Multiply $2$ by $1$ .

$2 \sec (θ) \tan (θ) + 2 \sec (θ) \tan (θ) \tan^{2} (θ) = 0$

Step 2.5

Multiply $\tan (θ)$ by $\tan^{2} (θ)$ by adding the exponents.

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Step 2.5.1

Move $\tan^{2} (θ)$ .

$2 \sec (θ) \tan (θ) + 2 \sec (θ) (\tan^{2} (θ) \tan (θ)) = 0$

Step 2.5.2

Multiply $\tan^{2} (θ)$ by $\tan (θ)$ .

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Step 2.5.2.1

Raise $\tan (θ)$ to the power of $1$ .

$2 \sec (θ) \tan (θ) + 2 \sec (θ) (\tan^{2} (θ) \tan (θ)) = 0$

Step 2.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 \sec (θ) \tan (θ) + 2 \sec (θ) {\tan (θ)}^{2 + 1} = 0$

Step 2.5.3

Add $2$ and $1$ .

$2 \sec (θ) \tan (θ) + 2 \sec (θ) \tan^{3} (θ) = 0$

Step 2.6

Reorder the polynomial.

$2 \sec (θ) \tan^{3} (θ) + 2 \sec (θ) \tan (θ) = 0$

Step 2.7

Simplify the left side.

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Step 2.7.1

Simplify $2 \sec (θ) \tan^{3} (θ) + 2 \sec (θ) \tan (θ)$ .

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Step 2.7.1.1

Factor $2 \sec (θ) \tan (θ)$ out of $2 \sec (θ) \tan^{3} (θ) + 2 \sec (θ) \tan (θ)$ .

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Step 2.7.1.1.1

Factor $2 \sec (θ) \tan (θ)$ out of $2 \sec (θ) \tan^{3} (θ)$ .

$2 \sec (θ) \tan (θ) (\tan^{2} (θ)) + 2 \sec (θ) \tan (θ) = 0$

Step 2.7.1.1.2

Factor $2 \sec (θ) \tan (θ)$ out of $2 \sec (θ) \tan (θ)$ .

$2 \sec (θ) \tan (θ) (\tan^{2} (θ)) + 2 \sec (θ) \tan (θ) (1) = 0$

Step 2.7.1.1.3

Factor $2 \sec (θ) \tan (θ)$ out of $2 \sec (θ) \tan (θ) (\tan^{2} (θ)) + 2 \sec (θ) \tan (θ) (1)$ .

$2 \sec (θ) \tan (θ) (\tan^{2} (θ) + 1) = 0$

Step 2.7.1.2

Apply pythagorean identity.

$2 \sec (θ) \tan (θ) \sec^{2} (θ) = 0$

Step 2.7.1.3

Multiply $\sec (θ)$ by $\sec^{2} (θ)$ by adding the exponents.

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Step 2.7.1.3.1

Move $\sec^{2} (θ)$ .

$2 (\sec^{2} (θ) \sec (θ)) \tan (θ) = 0$

Step 2.7.1.3.2

Multiply $\sec^{2} (θ)$ by $\sec (θ)$ .

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Step 2.7.1.3.2.1

Raise $\sec (θ)$ to the power of $1$ .

$2 (\sec^{2} (θ) \sec^{1} (θ)) \tan (θ) = 0$

Step 2.7.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 {\sec (θ)}^{2 + 1} \tan (θ) = 0$

Step 2.7.1.3.3

Add $2$ and $1$ .

$2 \sec^{3} (θ) \tan (θ) = 0$

Step 2.8

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

$\sec^{3} (θ) = 0$

$\tan (θ) = 0$

Step 2.9

Set $\sec^{3} (θ)$ equal to $0$ and solve for $θ$ .

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Step 2.9.1

Set $\sec^{3} (θ)$ equal to $0$ .

$\sec^{3} (θ) = 0$

Step 2.9.2

Solve $\sec^{3} (θ) = 0$ for $θ$ .

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Step 2.9.2.1

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

$\sec (θ) = \sqrt[3]{0}$

Step 2.9.2.2

Simplify $\sqrt[3]{0}$ .

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Step 2.9.2.2.1

Rewrite $0$ as $0^{3}$ .

$\sec (θ) = \sqrt[3]{0^{3}}$

Step 2.9.2.2.2

Pull terms out from under the radical, assuming real numbers.

$\sec (θ) = 0$

Step 2.9.2.3

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 2.10

Set $\tan (θ)$ equal to $0$ and solve for $θ$ .

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Step 2.10.1

Set $\tan (θ)$ equal to $0$ .

$\tan (θ) = 0$

Step 2.10.2

Solve $\tan (θ) = 0$ for $θ$ .

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Step 2.10.2.1

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

$θ = \arctan (0)$

Step 2.10.2.2

Simplify the right side.

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Step 2.10.2.2.1

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

$θ = 0$

Step 2.10.2.3

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.

$θ = π + 0$

Step 2.10.2.4

Add $π$ and $0$ .

$θ = π$

Step 2.10.2.5

Find the period of $\tan (θ)$ .

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Step 2.10.2.5.1

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

$\frac{π}{| b |}$

Step 2.10.2.5.2

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

$\frac{π}{| 1 |}$

Step 2.10.2.5.3

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

$\frac{π}{1}$

Step 2.10.2.5.4

Divide $π$ by $1$ .

$π$

Step 2.10.2.6

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

$θ = π n, π + π n$ , for any integer $n$

Step 2.11

The final solution is all the values that make $2 \sec^{3} (θ) \tan (θ) = 0$ true.

$θ = π n, π + π n$ , for any integer $n$

Step 2.12

Consolidate the answers.

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

$θ = \frac{π}{2} + π n$ , 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 $0$ .

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

Step 4

Evaluate $2 \sec (θ) + \tan (θ)$ at each $θ$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $θ = 0$ .

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Step 4.1.1

Substitute $0$ for $θ$ .

$2 \sec (0) + \tan (0)$

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

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

$2 \cdot 1 + \tan (0)$

Step 4.1.2.1.2

Multiply $2$ by $1$ .

$2 + \tan (0)$

Step 4.1.2.1.3

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

$2 + 0$

Step 4.1.2.2

Add $2$ and $0$ .

$2$

Step 4.2

Evaluate at $θ = π$ .

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Step 4.2.1

Substitute $π$ for $θ$ .

$2 \sec (π) + \tan (π)$

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 secant is negative in the second quadrant.

$2 (- \sec (0)) + \tan (π)$

Step 4.2.2.1.2

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

$2 (- 1 \cdot 1) + \tan (π)$

Step 4.2.2.1.3

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

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Step 4.2.2.1.3.1

Multiply $- 1$ by $1$ .

$2 \cdot - 1 + \tan (π)$

Step 4.2.2.1.3.2

Multiply $2$ by $- 1$ .

$- 2 + \tan (π)$

Step 4.2.2.1.4

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.

$- 2 - \tan (0)$

Step 4.2.2.1.5

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

$- 2 - 0$

Step 4.2.2.1.6

Multiply $- 1$ by $0$ .

$- 2 + 0$

Step 4.2.2.2

Add $- 2$ and $0$ .

$- 2$

Step 4.3

List all of the points.

$(0 + 2 π n, 2), (π + 2 π n, - 2)$ , for any integer $n$

Step 5