Calculus Examples

$10 \sec (x) + 5 \tan (x)$

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 $10 \sec (x) + 5 \tan (x)$ with respect to $x$ is $\frac{d}{d x} [10 \sec (x)] + \frac{d}{d x} [5 \tan (x)]$ .

$\frac{d}{d x} [10 \sec (x)] + \frac{d}{d x} [5 \tan (x)]$

Step 1.1.2

Evaluate $\frac{d}{d x} [10 \sec (x)]$ .

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Step 1.1.2.1

Since $10$ is constant with respect to $x$ , the derivative of $10 \sec (x)$ with respect to $x$ is $10 \frac{d}{d x} [\sec (x)]$ .

$10 \frac{d}{d x} [\sec (x)] + \frac{d}{d x} [5 \tan (x)]$

Step 1.1.2.2

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

$10 \sec (x) \tan (x) + \frac{d}{d x} [5 \tan (x)]$

Step 1.1.3

Evaluate $\frac{d}{d x} [5 \tan (x)]$ .

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Step 1.1.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \tan (x)$ with respect to $x$ is $5 \frac{d}{d x} [\tan (x)]$ .

$10 \sec (x) \tan (x) + 5 \frac{d}{d x} [\tan (x)]$

Step 1.1.3.2

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

$f' (x) = 10 \sec (x) \tan (x) + 5 \sec^{2} (x)$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $10 \sec (x) \tan (x) + 5 \sec^{2} (x)$ .

$10 \sec (x) \tan (x) + 5 \sec^{2} (x)$

Step 2

Set the first derivative equal to $0$ then solve the equation $10 \sec (x) \tan (x) + 5 \sec^{2} (x) = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$10 \sec (x) \tan (x) + 5 \sec^{2} (x) = 0$

Step 2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , 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 $\sec (x)$ 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 $10 \sec (x) + 5 \tan (x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{7 π}{6}$ .

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Step 4.1.1

Substitute $\frac{7 π}{6}$ for $x$ .

$10 \sec (\frac{7 π}{6}) + 5 \tan (\frac{7 π}{6})$

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. Make the expression negative because secant is negative in the third quadrant.

$10 (- \sec (\frac{π}{6})) + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.2

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

$10 (- \frac{2}{\sqrt{3}}) + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.3

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$10 (- (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})) + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.4

Combine and simplify the denominator.

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Step 4.1.2.1.4.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$10 (- \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}) + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$10 (- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}) + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$10 (- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}) + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.4.4

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

$10 (- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}) + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.4.5

Add $1$ and $1$ .

$10 (- \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}) + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.1.2.1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$10 (- \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}) + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$10 (- \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}) + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$10 (- \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}) + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.4.6.4

Cancel the common factor of $2$ .

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Step 4.1.2.1.4.6.4.1

Cancel the common factor.

$10 (- \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}) + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.4.6.4.2

Rewrite the expression.

$10 (- \frac{2 \sqrt{3}}{3^{1}}) + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.4.6.5

Evaluate the exponent.

$10 (- \frac{2 \sqrt{3}}{3}) + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.5

Multiply $10 (- \frac{2 \sqrt{3}}{3})$ .

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Step 4.1.2.1.5.1

Multiply $- 1$ by $10$ .

$- 10 \frac{2 \sqrt{3}}{3} + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.5.2

Combine $- 10$ and $\frac{2 \sqrt{3}}{3}$ .

$\frac{- 10 (2 \sqrt{3})}{3} + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.5.3

Multiply $2$ by $- 10$ .

$\frac{- 20 \sqrt{3}}{3} + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.6

Move the negative in front of the fraction.

$- \frac{20 \sqrt{3}}{3} + 5 \tan (\frac{7 π}{6})$

Step 4.1.2.1.7

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

$- \frac{20 \sqrt{3}}{3} + 5 \tan (\frac{π}{6})$

Step 4.1.2.1.8

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$- \frac{20 \sqrt{3}}{3} + 5 \frac{\sqrt{3}}{3}$

Step 4.1.2.1.9

Combine $5$ and $\frac{\sqrt{3}}{3}$ .

$- \frac{20 \sqrt{3}}{3} + \frac{5 \sqrt{3}}{3}$

Step 4.1.2.2

Simplify terms.

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Step 4.1.2.2.1

Combine the numerators over the common denominator.

$\frac{- 20 \sqrt{3} + 5 \sqrt{3}}{3}$

Step 4.1.2.2.2

Add $- 20 \sqrt{3}$ and $5 \sqrt{3}$ .

$\frac{- 15 \sqrt{3}}{3}$

Step 4.1.2.2.3

Cancel the common factor of $- 15$ and $3$ .

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Step 4.1.2.2.3.1

Factor $3$ out of $- 15 \sqrt{3}$ .

$\frac{3 (- 5 \sqrt{3})}{3}$

Step 4.1.2.2.3.2

Cancel the common factors.

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Step 4.1.2.2.3.2.1

Factor $3$ out of $3$ .

$\frac{3 (- 5 \sqrt{3})}{3 (1)}$

Step 4.1.2.2.3.2.2

Cancel the common factor.

$\frac{3 (- 5 \sqrt{3})}{3 \cdot 1}$

Step 4.1.2.2.3.2.3

Rewrite the expression.

$\frac{- 5 \sqrt{3}}{1}$

Step 4.1.2.2.3.2.4

Divide $- 5 \sqrt{3}$ by $1$ .

$- 5 \sqrt{3}$

Step 4.2

Evaluate at $x = \frac{11 π}{6}$ .

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Step 4.2.1

Substitute $\frac{11 π}{6}$ for $x$ .

$10 \sec (\frac{11 π}{6}) + 5 \tan (\frac{11 π}{6})$

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.

$10 \sec (\frac{π}{6}) + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.2

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

$10 \frac{2}{\sqrt{3}} + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.3

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$10 (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.4

Combine and simplify the denominator.

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Step 4.2.2.1.4.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$10 \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$10 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$10 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.4.4

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

$10 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.4.5

Add $1$ and $1$ .

$10 \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.2.2.1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$10 \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$10 \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$10 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.4.6.4

Cancel the common factor of $2$ .

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Step 4.2.2.1.4.6.4.1

Cancel the common factor.

$10 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.4.6.4.2

Rewrite the expression.

$10 \frac{2 \sqrt{3}}{3^{1}} + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.4.6.5

Evaluate the exponent.

$10 \frac{2 \sqrt{3}}{3} + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.5

Multiply $10 \frac{2 \sqrt{3}}{3}$ .

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Step 4.2.2.1.5.1

Combine $10$ and $\frac{2 \sqrt{3}}{3}$ .

$\frac{10 (2 \sqrt{3})}{3} + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.5.2

Multiply $2$ by $10$ .

$\frac{20 \sqrt{3}}{3} + 5 \tan (\frac{11 π}{6})$

Step 4.2.2.1.6

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.

$\frac{20 \sqrt{3}}{3} + 5 (- \tan (\frac{π}{6}))$

Step 4.2.2.1.7

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{20 \sqrt{3}}{3} + 5 (- \frac{\sqrt{3}}{3})$

Step 4.2.2.1.8

Multiply $5 (- \frac{\sqrt{3}}{3})$ .

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Step 4.2.2.1.8.1

Multiply $- 1$ by $5$ .

$\frac{20 \sqrt{3}}{3} - 5 \frac{\sqrt{3}}{3}$

Step 4.2.2.1.8.2

Combine $- 5$ and $\frac{\sqrt{3}}{3}$ .

$\frac{20 \sqrt{3}}{3} + \frac{- 5 \sqrt{3}}{3}$

Step 4.2.2.1.9

Move the negative in front of the fraction.

$\frac{20 \sqrt{3}}{3} - \frac{5 \sqrt{3}}{3}$

Step 4.2.2.2

Simplify terms.

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Step 4.2.2.2.1

Combine the numerators over the common denominator.

$\frac{20 \sqrt{3} - 5 \sqrt{3}}{3}$

Step 4.2.2.2.2

Subtract $5 \sqrt{3}$ from $20 \sqrt{3}$ .

$\frac{15 \sqrt{3}}{3}$

Step 4.2.2.2.3

Cancel the common factor of $15$ and $3$ .

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Step 4.2.2.2.3.1

Factor $3$ out of $15 \sqrt{3}$ .

$\frac{3 (5 \sqrt{3})}{3}$

Step 4.2.2.2.3.2

Cancel the common factors.

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Step 4.2.2.2.3.2.1

Factor $3$ out of $3$ .

$\frac{3 (5 \sqrt{3})}{3 (1)}$

Step 4.2.2.2.3.2.2

Cancel the common factor.

$\frac{3 (5 \sqrt{3})}{3 \cdot 1}$

Step 4.2.2.2.3.2.3

Rewrite the expression.

$\frac{5 \sqrt{3}}{1}$

Step 4.2.2.2.3.2.4

Divide $5 \sqrt{3}$ by $1$ .

$5 \sqrt{3}$

Step 4.3

List all of the points.

$(\frac{7 π}{6} + 2 π n, - 5 \sqrt{3}), (\frac{11 π}{6} + 2 π n, 5 \sqrt{3})$ , for any integer $n$

Step 5