Calculus Examples

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

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

By the Sum Rule, the derivative of $\sin (x) - \sqrt{3} \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- \sqrt{3} \cos (x)]$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

Tap for more steps...

Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Multiply $- 1$ by $- 1$ .

$\cos (x) + 1 \sqrt{3} \sin (x)$

Step 1.1.3.4

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

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

Step 1.2

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

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

Step 2

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

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

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

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Rewrite the expression.

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

Step 2.4

Separate fractions.

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Separate fractions.

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

Step 2.8

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

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

Step 2.9

Divide $0$ by $1$ .

$1 + \sqrt{3} \tan (x) = 0 \sec (x)$

Step 2.10

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

$1 + \sqrt{3} \tan (x) = 0$

Step 2.11

Subtract $1$ from both sides of the equation.

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

Step 2.12

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

Tap for more steps...

Step 2.12.1

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

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

Step 2.12.2

Simplify the left side.

Tap for more steps...

Step 2.12.2.1

Cancel the common factor of $\sqrt{3}$ .

Tap for more steps...

Step 2.12.2.1.1

Cancel the common factor.

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

Step 2.12.2.1.2

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

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

Step 2.12.3

Simplify the right side.

Tap for more steps...

Step 2.12.3.1

Move the negative in front of the fraction.

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

Step 2.12.3.2

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 2.12.3.3

Combine and simplify the denominator.

Tap for more steps...

Step 2.12.3.3.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 2.12.3.3.2

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

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

Step 2.12.3.3.3

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

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

Step 2.12.3.3.4

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

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

Step 2.12.3.3.5

Add $1$ and $1$ .

$\tan (x) = - \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.12.3.3.6

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

Tap for more steps...

Step 2.12.3.3.6.1

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

$\tan (x) = - \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.12.3.3.6.2

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

$\tan (x) = - \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.12.3.3.6.3

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

$\tan (x) = - \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.12.3.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.12.3.3.6.4.1

Cancel the common factor.

$\tan (x) = - \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.12.3.3.6.4.2

Rewrite the expression.

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

Step 2.12.3.3.6.5

Evaluate the exponent.

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

Step 2.13

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

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

Step 2.14

Simplify the right side.

Tap for more steps...

Step 2.14.1

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

$x = - \frac{π}{6}$

Step 2.15

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

$x = - \frac{π}{6} - π$

Step 2.16

Simplify the expression to find the second solution.

Tap for more steps...

Step 2.16.1

Add $2 π$ to $- \frac{π}{6} - π$ .

$x = - \frac{π}{6} - π + 2 π$

Step 2.16.2

The resulting angle of $\frac{5 π}{6}$ is positive and coterminal with $- \frac{π}{6} - π$ .

$x = \frac{5 π}{6}$

Step 2.17

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

Tap for more steps...

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

Add $π$ to every negative angle to get positive angles.

Tap for more steps...

Step 2.18.1

Add $π$ to $- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + π$

Step 2.18.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$π \cdot \frac{6}{6} - \frac{π}{6}$

Step 2.18.3

Combine fractions.

Tap for more steps...

Step 2.18.3.1

Combine $π$ and $\frac{6}{6}$ .

$\frac{π \cdot 6}{6} - \frac{π}{6}$

Step 2.18.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 6 - π}{6}$

Step 2.18.4

Simplify the numerator.

Tap for more steps...

Step 2.18.4.1

Move $6$ to the left of $π$ .

$\frac{6 \cdot π - π}{6}$

Step 2.18.4.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{6}$

Step 2.18.5

List the new angles.

$x = \frac{5 π}{6}$

Step 2.19

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

$x = \frac{5 π}{6} + π n, \frac{5 π}{6} + π n$ , for any integer $n$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

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

Tap for more steps...

Step 4.1

Evaluate at $x = \frac{5 π}{6}$ .

Tap for more steps...

Step 4.1.1

Substitute $\frac{5 π}{6}$ for $x$ .

$\sin (\frac{5 π}{6}) - \sqrt{3} \cos (\frac{5 π}{6})$

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

Simplify each term.

Tap for more steps...

Step 4.1.2.1.1

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

$\sin (\frac{π}{6}) - \sqrt{3} \cos (\frac{5 π}{6})$

Step 4.1.2.1.2

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{1}{2} - \sqrt{3} \cos (\frac{5 π}{6})$

Step 4.1.2.1.3

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 second quadrant.

$\frac{1}{2} - \sqrt{3} (- \cos (\frac{π}{6}))$

Step 4.1.2.1.4

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

$\frac{1}{2} - \sqrt{3} (- \frac{\sqrt{3}}{2})$

Step 4.1.2.1.5

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

Tap for more steps...

Step 4.1.2.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.2.1.5.2

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

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

Step 4.1.2.1.5.3

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

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

Step 4.1.2.1.5.4

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

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

Step 4.1.2.1.5.5

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

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

Step 4.1.2.1.5.6

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

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

Step 4.1.2.1.5.7

Add $1$ and $1$ .

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

Step 4.1.2.1.6

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

Tap for more steps...

Step 4.1.2.1.6.1

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

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

Step 4.1.2.1.6.2

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

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

Step 4.1.2.1.6.3

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

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

Step 4.1.2.1.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.2.1.6.4.1

Cancel the common factor.

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

Step 4.1.2.1.6.4.2

Rewrite the expression.

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

Step 4.1.2.1.6.5

Evaluate the exponent.

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

Step 4.1.2.2

Combine fractions.

Tap for more steps...

Step 4.1.2.2.1

Combine the numerators over the common denominator.

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

Step 4.1.2.2.2

Simplify the expression.

Tap for more steps...

Step 4.1.2.2.2.1

Add $1$ and $3$ .

$\frac{4}{2}$

Step 4.1.2.2.2.2

Divide $4$ by $2$ .

$2$

Step 4.2

Evaluate at $x = \frac{11 π}{6}$ .

Tap for more steps...

Step 4.2.1

Substitute $\frac{11 π}{6}$ for $x$ .

$\sin (\frac{11 π}{6}) - \sqrt{3} \cos (\frac{11 π}{6})$

Step 4.2.2

Simplify.

Tap for more steps...

Step 4.2.2.1

Simplify each term.

Tap for more steps...

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 fourth quadrant.

$- \sin (\frac{π}{6}) - \sqrt{3} \cos (\frac{11 π}{6})$

Step 4.2.2.1.2

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$- \frac{1}{2} - \sqrt{3} \cos (\frac{11 π}{6})$

Step 4.2.2.1.3

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

$- \frac{1}{2} - \sqrt{3} \cos (\frac{π}{6})$

Step 4.2.2.1.4

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

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

Step 4.2.2.1.5

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

Tap for more steps...

Step 4.2.2.1.5.1

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

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

Step 4.2.2.1.5.2

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

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

Step 4.2.2.1.5.3

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

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

Step 4.2.2.1.5.4

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

$- \frac{1}{2} - \frac{{\sqrt{3}}^{1 + 1}}{2}$

Step 4.2.2.1.5.5

Add $1$ and $1$ .

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

Step 4.2.2.1.6

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

Tap for more steps...

Step 4.2.2.1.6.1

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

$- \frac{1}{2} - \frac{{(3^{\frac{1}{2}})}^{2}}{2}$

Step 4.2.2.1.6.2

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

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

Step 4.2.2.1.6.3

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

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

Step 4.2.2.1.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.2.1.6.4.1

Cancel the common factor.

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

Step 4.2.2.1.6.4.2

Rewrite the expression.

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

Step 4.2.2.1.6.5

Evaluate the exponent.

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

Step 4.2.2.2

Combine fractions.

Tap for more steps...

Step 4.2.2.2.1

Combine the numerators over the common denominator.

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

Step 4.2.2.2.2

Simplify the expression.

Tap for more steps...

Step 4.2.2.2.2.1

Subtract $3$ from $- 1$ .

$\frac{- 4}{2}$

Step 4.2.2.2.2.2

Divide $- 4$ by $2$ .

$- 2$

Step 4.3

List all of the points.

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

Step 5