Calculus Examples

10 sec (x) + 5 tan (x)

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

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)

$10 \sec (x) + 5 \tan (x)$ with respect to

x

$x$ is

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

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

\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)]

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

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Step 1.1.2.1

Since

10

$10$ is constant with respect to

x

$x$ , the derivative of

10 sec (x)

$10 \sec (x)$ with respect to

x

$x$ is

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

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

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

$\sec (x)$ with respect to

x

$x$ is

sec (x) tan (x)

$\sec (x) \tan (x)$ .

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

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

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

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

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Step 1.1.3.1

Since

5

$5$ is constant with respect to

x

$x$ , the derivative of

5 tan (x)

$5 \tan (x)$ with respect to

x

$x$ is

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

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

10 sec (x) tan (x) + 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)

$\tan (x)$ with respect to

x

$x$ is

{sec}^{2} (x)

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

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

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

$n$

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , 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

$n$

${x | x = \frac{π}{2} + π n}$

$n$ , 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

$n$

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

$n$

Step 5