Calculus Examples

$f (x) = \sqrt{3} \sin (x) + \cos (x)$

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 $\sqrt{3} \sin (x) + \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\sqrt{3} \sin (x)] + \frac{d}{d x} [\cos (x)]$ .

$\frac{d}{d x} [\sqrt{3} \sin (x)] + \frac{d}{d x} [\cos (x)]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\sqrt{3} \sin (x)]$ .

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Step 1.1.2.1

Since $\sqrt{3}$ is constant with respect to $x$ , the derivative of $\sqrt{3} \sin (x)$ with respect to $x$ is $\sqrt{3} \frac{d}{d x} [\sin (x)]$ .

$\sqrt{3} \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\cos (x)]$

Step 1.1.2.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\sqrt{3} \cos (x) + \frac{d}{d x} [\cos (x)]$

Step 1.1.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$f' (x) = \sqrt{3} \cos (x) - \sin (x)$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\sqrt{3} \cos (x) - \sin (x)$ .

$\sqrt{3} \cos (x) - \sin (x)$

Step 2

Set the first derivative equal to $0$ then solve the equation $\sqrt{3} \cos (x) - \sin (x) = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\sqrt{3} \cos (x) - \sin (x) = 0$

Step 2.2

Divide each term in the equation by $\cos (x)$ .

$\frac{\sqrt{3} \cos (x)}{\cos (x)} + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 2.3

Cancel the common factor of $\cos (x)$ .

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Step 2.3.1

Cancel the common factor.

$\frac{\sqrt{3} \cos (x)}{\cos (x)} + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 2.3.2

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

$\sqrt{3} + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 2.4

Separate fractions.

$\sqrt{3} + \frac{- 1}{1} \cdot \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 2.5

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\sqrt{3} + \frac{- 1}{1} \cdot \tan (x) = \frac{0}{\cos (x)}$

Step 2.6

Divide $- 1$ by $1$ .

$\sqrt{3} - \tan (x) = \frac{0}{\cos (x)}$

Step 2.7

Separate fractions.

$\sqrt{3} - \tan (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 2.8

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sqrt{3} - \tan (x) = \frac{0}{1} \cdot \sec (x)$

Step 2.9

Divide $0$ by $1$ .

$\sqrt{3} - \tan (x) = 0 \sec (x)$

Step 2.10

Multiply $0$ by $\sec (x)$ .

$\sqrt{3} - \tan (x) = 0$

Step 2.11

Subtract $\sqrt{3}$ from both sides of the equation.

$- \tan (x) = - \sqrt{3}$

Step 2.12

Divide each term in $- \tan (x) = - \sqrt{3}$ by $- 1$ and simplify.

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Step 2.12.1

Divide each term in $- \tan (x) = - \sqrt{3}$ by $- 1$ .

$\frac{- \tan (x)}{- 1} = \frac{- \sqrt{3}}{- 1}$

Step 2.12.2

Simplify the left side.

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Step 2.12.2.1

Dividing two negative values results in a positive value.

$\frac{\tan (x)}{1} = \frac{- \sqrt{3}}{- 1}$

Step 2.12.2.2

Divide $\tan (x)$ by $1$ .

$\tan (x) = \frac{- \sqrt{3}}{- 1}$

Step 2.12.3

Simplify the right side.

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Step 2.12.3.1

Dividing two negative values results in a positive value.

$\tan (x) = \frac{\sqrt{3}}{1}$

Step 2.12.3.2

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

$\tan (x) = \sqrt{3}$

Step 2.13

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

$x = \arctan (\sqrt{3})$

Step 2.14

Simplify the right side.

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Step 2.14.1

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

$x = \frac{π}{3}$

Step 2.15

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.

$x = π + \frac{π}{3}$

Step 2.16

Simplify $π + \frac{π}{3}$ .

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Step 2.16.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = π \cdot \frac{3}{3} + \frac{π}{3}$

Step 2.16.2

Combine fractions.

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Step 2.16.2.1

Combine $π$ and $\frac{3}{3}$ .

$x = \frac{π \cdot 3}{3} + \frac{π}{3}$

Step 2.16.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 + π}{3}$

Step 2.16.3

Simplify the numerator.

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Step 2.16.3.1

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π + π}{3}$

Step 2.16.3.2

Add $3 π$ and $π$ .

$x = \frac{4 π}{3}$

Step 2.17

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

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Step 2.17.1

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

$\frac{π}{| b |}$

Step 2.17.2

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

$\frac{π}{| 1 |}$

Step 2.17.3

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

$\frac{π}{1}$

Step 2.17.4

Divide $π$ by $1$ .

$π$

Step 2.18

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

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

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $\sqrt{3} \sin (x) + \cos (x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{π}{3}$ .

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Step 4.1.1

Substitute $\frac{π}{3}$ for $x$ .

$\sqrt{3} \sin (\frac{π}{3}) + \cos (\frac{π}{3})$

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 $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\sqrt{3} \frac{\sqrt{3}}{2} + \cos (\frac{π}{3})$

Step 4.1.2.1.2

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

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Step 4.1.2.1.2.1

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

$\frac{\sqrt{3} \sqrt{3}}{2} + \cos (\frac{π}{3})$

Step 4.1.2.1.2.2

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

$\frac{{\sqrt{3}}^{1} \sqrt{3}}{2} + \cos (\frac{π}{3})$

Step 4.1.2.1.2.3

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

$\frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{2} + \cos (\frac{π}{3})$

Step 4.1.2.1.2.4

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

$\frac{{\sqrt{3}}^{1 + 1}}{2} + \cos (\frac{π}{3})$

Step 4.1.2.1.2.5

Add $1$ and $1$ .

$\frac{{\sqrt{3}}^{2}}{2} + \cos (\frac{π}{3})$

Step 4.1.2.1.3

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

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Step 4.1.2.1.3.1

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

$\frac{{(3^{\frac{1}{2}})}^{2}}{2} + \cos (\frac{π}{3})$

Step 4.1.2.1.3.2

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

$\frac{3^{\frac{1}{2} \cdot 2}}{2} + \cos (\frac{π}{3})$

Step 4.1.2.1.3.3

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

$\frac{3^{\frac{2}{2}}}{2} + \cos (\frac{π}{3})$

Step 4.1.2.1.3.4

Cancel the common factor of $2$ .

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Step 4.1.2.1.3.4.1

Cancel the common factor.

$\frac{3^{\frac{2}{2}}}{2} + \cos (\frac{π}{3})$

Step 4.1.2.1.3.4.2

Rewrite the expression.

$\frac{3^{1}}{2} + \cos (\frac{π}{3})$

Step 4.1.2.1.3.5

Evaluate the exponent.

$\frac{3}{2} + \cos (\frac{π}{3})$

Step 4.1.2.1.4

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{3}{2} + \frac{1}{2}$

Step 4.1.2.2

Combine fractions.

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Step 4.1.2.2.1

Combine the numerators over the common denominator.

$\frac{3 + 1}{2}$

Step 4.1.2.2.2

Simplify the expression.

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Step 4.1.2.2.2.1

Add $3$ and $1$ .

$\frac{4}{2}$

Step 4.1.2.2.2.2

Divide $4$ by $2$ .

$2$

Step 4.2

Evaluate at $x = \frac{4 π}{3}$ .

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Step 4.2.1

Substitute $\frac{4 π}{3}$ for $x$ .

$\sqrt{3} \sin (\frac{4 π}{3}) + \cos (\frac{4 π}{3})$

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 sine is negative in the third quadrant.

$\sqrt{3} (- \sin (\frac{π}{3})) + \cos (\frac{4 π}{3})$

Step 4.2.2.1.2

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

$\sqrt{3} (- \frac{\sqrt{3}}{2}) + \cos (\frac{4 π}{3})$

Step 4.2.2.1.3

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

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Step 4.2.2.1.3.1

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

$- \frac{\sqrt{3} \sqrt{3}}{2} + \cos (\frac{4 π}{3})$

Step 4.2.2.1.3.2

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

$- \frac{{\sqrt{3}}^{1} \sqrt{3}}{2} + \cos (\frac{4 π}{3})$

Step 4.2.2.1.3.3

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

$- \frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{2} + \cos (\frac{4 π}{3})$

Step 4.2.2.1.3.4

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

$- \frac{{\sqrt{3}}^{1 + 1}}{2} + \cos (\frac{4 π}{3})$

Step 4.2.2.1.3.5

Add $1$ and $1$ .

$- \frac{{\sqrt{3}}^{2}}{2} + \cos (\frac{4 π}{3})$

Step 4.2.2.1.4

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

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Step 4.2.2.1.4.1

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

$- \frac{{(3^{\frac{1}{2}})}^{2}}{2} + \cos (\frac{4 π}{3})$

Step 4.2.2.1.4.2

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

$- \frac{3^{\frac{1}{2} \cdot 2}}{2} + \cos (\frac{4 π}{3})$

Step 4.2.2.1.4.3

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

$- \frac{3^{\frac{2}{2}}}{2} + \cos (\frac{4 π}{3})$

Step 4.2.2.1.4.4

Cancel the common factor of $2$ .

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Step 4.2.2.1.4.4.1

Cancel the common factor.

$- \frac{3^{\frac{2}{2}}}{2} + \cos (\frac{4 π}{3})$

Step 4.2.2.1.4.4.2

Rewrite the expression.

$- \frac{3^{1}}{2} + \cos (\frac{4 π}{3})$

Step 4.2.2.1.4.5

Evaluate the exponent.

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

Step 4.2.2.1.5

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

$- \frac{3}{2} - \cos (\frac{π}{3})$

Step 4.2.2.1.6

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$- \frac{3}{2} - \frac{1}{2}$

Step 4.2.2.2

Combine fractions.

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Step 4.2.2.2.1

Combine the numerators over the common denominator.

$\frac{- 3 - 1}{2}$

Step 4.2.2.2.2

Simplify the expression.

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Step 4.2.2.2.2.1

Subtract $1$ from $- 3$ .

$\frac{- 4}{2}$

Step 4.2.2.2.2.2

Divide $- 4$ by $2$ .

$- 2$

Step 4.3

List all of the points.

$(\frac{π}{3} + 2 π n, 2), (\frac{4 π}{3} + 2 π n, - 2)$ , for any integer $n$

Step 5