Calculus Examples

$y = \sin (x)$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

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

$f' (x) = \cos (x)$

Step 1.2

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

$\cos (x)$

Step 2

Set the first derivative equal to $0$ then solve the equation $\cos (x) = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$\cos (x) = 0$

Step 2.2

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

$x = \arccos (0)$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 2.4

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{2}$

Step 2.5

Simplify $2 π - \frac{π}{2}$ .

Tap for more steps...

Step 2.5.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 2.5.2

Combine fractions.

Tap for more steps...

Step 2.5.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 2.5.3

Simplify the numerator.

Tap for more steps...

Step 2.5.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 2.5.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 2.6

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

Tap for more steps...

Step 2.6.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 2.6.2

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

$\frac{2 π}{| 1 |}$

Step 2.6.3

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

$\frac{2 π}{1}$

Step 2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.7

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 2.8

Consolidate the answers.

$x = \frac{π}{2} + π n$ , 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 $\sin (x)$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = \frac{π}{2}$ .

Tap for more steps...

Step 4.1.1

Substitute $\frac{π}{2}$ for $x$ .

$\sin (\frac{π}{2})$

Step 4.1.2

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

$1$

Step 4.2

Evaluate at $x = \frac{3 π}{2}$ .

Tap for more steps...

Step 4.2.1

Substitute $\frac{3 π}{2}$ for $x$ .

$\sin (\frac{3 π}{2})$

Step 4.2.2

Simplify.

Tap for more steps...

Step 4.2.2.1

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.

$- \sin (\frac{π}{2})$

Step 4.2.2.2

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

$- 1 \cdot 1$

Step 4.2.2.3

Multiply $- 1$ by $1$ .

$- 1$

Step 4.3

List all of the points.

$(\frac{π}{2} + 2 π n, 1), (\frac{3 π}{2} + 2 π n, - 1)$ , for any integer $n$

Step 5