Calculus Examples

$y = x - \sin (x)$

Step 1

Write $y = x - \sin (x)$ as a function.

$f (x) = x - \sin (x)$

Step 2

Find the first derivative of the function.

Tap for more steps...

Step 2.1

Differentiate.

Tap for more steps...

Step 2.1.1

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

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

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- \sin (x)]$

Step 2.2

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

Tap for more steps...

Step 2.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (x)$ with respect to $x$ is $- \frac{d}{d x} [\sin (x)]$ .

$1 - \frac{d}{d x} [\sin (x)]$

Step 2.2.2

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

$1 - \cos (x)$

Step 3

Find the second derivative of the function.

Tap for more steps...

Step 3.1

Differentiate.

Tap for more steps...

Step 3.1.1

By the Sum Rule, the derivative of $1 - \cos (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (x)]$ .

$f'' (x) = \frac{d}{d x} (1) + \frac{d}{d x} (- \cos (x))$

Step 3.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f'' (x) = 0 + \frac{d}{d x} (- \cos (x))$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (x)]$ .

$f'' (x) = 0 - \frac{d}{d x} \cos (x)$

Step 3.2.2

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

$f'' (x) = 0 + \sin (x)$

Step 3.2.3

Multiply $- 1$ by $- 1$ .

$f'' (x) = 0 + 1 \sin (x)$

Step 3.2.4

Multiply $\sin (x)$ by $1$ .

$f'' (x) = 0 + \sin (x)$

Step 3.3

Add $0$ and $\sin (x)$ .

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

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$1 - \cos (x) = 0$

Step 5

Subtract $1$ from both sides of the equation.

$- \cos (x) = - 1$

Step 6

Divide each term in $- \cos (x) = - 1$ by $- 1$ and simplify.

Tap for more steps...

Step 6.1

Divide each term in $- \cos (x) = - 1$ by $- 1$ .

$\frac{- \cos (x)}{- 1} = \frac{- 1}{- 1}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{\cos (x)}{1} = \frac{- 1}{- 1}$

Step 6.2.2

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

$\cos (x) = \frac{- 1}{- 1}$

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Divide $- 1$ by $- 1$ .

$\cos (x) = 1$

Step 7

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

$x = \arccos (1)$

Step 8

Simplify the right side.

Tap for more steps...

Step 8.1

The exact value of $\arccos (1)$ is $0$ .

$x = 0$

Step 9

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.

$x = 2 π - 0$

Step 10

Subtract $0$ from $2 π$ .

$x = 2 π$

Step 11

The solution to the equation $x = 0$ .

$x = 0, 2 π$

Step 12

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\sin (0)$

Step 13

The exact value of $\sin (0)$ is $0$ .

$0$

Step 14

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

Tap for more steps...

Step 14.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, 0) \cup (0, 2 π) \cup (2 π, \infty)$

Step 14.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $1 - \cos (x)$ to check if the result is negative or positive.

Tap for more steps...

Step 14.2.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = 1 - \cos (- 2)$

Step 14.2.2

Simplify the result.

Tap for more steps...

Step 14.2.2.1

Simplify each term.

Tap for more steps...

Step 14.2.2.1.1

Evaluate $\cos (- 2)$ .

$f' (- 2) = 1 + 0.41614683$

Step 14.2.2.1.2

Multiply $- 1$ by $- 0.41614683$ .

$f' (- 2) = 1 + 0.41614683$

Step 14.2.2.2

Add $1$ and $0.41614683$ .

$f' (- 2) = 1.41614683$

Step 14.2.2.3

The final answer is $1.41614683$ .

$1.41614683$

Step 14.3

Substitute any number, such as $3$ , from the interval $(0, 2 π)$ in the first derivative $1 - \cos (x)$ to check if the result is negative or positive.

Tap for more steps...

Step 14.3.1

Replace the variable $x$ with $3$ in the expression.

$f' (3) = 1 - \cos (3)$

Step 14.3.2

Simplify the result.

Tap for more steps...

Step 14.3.2.1

Simplify each term.

Tap for more steps...

Step 14.3.2.1.1

Evaluate $\cos (3)$ .

$f' (3) = 1 + 0.98999249$

Step 14.3.2.1.2

Multiply $- 1$ by $- 0.98999249$ .

$f' (3) = 1 + 0.98999249$

Step 14.3.2.2

Add $1$ and $0.98999249$ .

$f' (3) = 1.98999249$

Step 14.3.2.3

The final answer is $1.98999249$ .

$1.98999249$

Step 14.4

Substitute any number, such as $9$ , from the interval $(2 π, \infty)$ in the first derivative $1 - \cos (x)$ to check if the result is negative or positive.

Tap for more steps...

Step 14.4.1

Replace the variable $x$ with $9$ in the expression.

$f' (9) = 1 - \cos (9)$

Step 14.4.2

Simplify the result.

Tap for more steps...

Step 14.4.2.1

Simplify each term.

Tap for more steps...

Step 14.4.2.1.1

Evaluate $\cos (9)$ .

$f' (9) = 1 + 0.91113026$

Step 14.4.2.1.2

Multiply $- 1$ by $- 0.91113026$ .

$f' (9) = 1 + 0.91113026$

Step 14.4.2.2

Add $1$ and $0.91113026$ .

$f' (9) = 1.91113026$

Step 14.4.2.3

The final answer is $1.91113026$ .

$1.91113026$

Step 14.5

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 14.6

No local maxima or minima found for $f (x) = x - \sin (x)$ .

No local maxima or minima

Step 15