Calculus Examples

$f (x) = 4 x + 4 \cos (x)$

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 $4 x + 4 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [4 \cos (x)]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [4 \cos (x)]$

Step 1.1.2

Evaluate $\frac{d}{d x} [4 x]$ .

Tap for more steps...

Step 1.1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x] + \frac{d}{d x} [4 \cos (x)]$

Step 1.1.2.2

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

$4 \cdot 1 + \frac{d}{d x} [4 \cos (x)]$

Step 1.1.2.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [4 \cos (x)]$

Step 1.1.3

Evaluate $\frac{d}{d x} [4 \cos (x)]$ .

Tap for more steps...

Step 1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 \cos (x)$ with respect to $x$ is $4 \frac{d}{d x} [\cos (x)]$ .

$4 + 4 \frac{d}{d x} [\cos (x)]$

Step 1.1.3.2

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

$4 + 4 (- \sin (x))$

Step 1.1.3.3

Multiply $- 1$ by $4$ .

$f' (x) = 4 - 4 \sin (x)$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $4 - 4 \sin (x)$ .

$4 - 4 \sin (x)$

Step 2

Set the first derivative equal to $0$ then solve the equation $4 - 4 \sin (x) = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$4 - 4 \sin (x) = 0$

Step 2.2

Subtract $4$ from both sides of the equation.

$- 4 \sin (x) = - 4$

Step 2.3

Divide each term in $- 4 \sin (x) = - 4$ by $- 4$ and simplify.

Tap for more steps...

Step 2.3.1

Divide each term in $- 4 \sin (x) = - 4$ by $- 4$ .

$\frac{- 4 \sin (x)}{- 4} = \frac{- 4}{- 4}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 4 \sin (x)}{- 4} = \frac{- 4}{- 4}$

Step 2.3.2.1.2

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

$\sin (x) = \frac{- 4}{- 4}$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Divide $- 4$ by $- 4$ .

$\sin (x) = 1$

Step 2.4

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

$x = \arcsin (1)$

Step 2.5

Simplify the right side.

Tap for more steps...

Step 2.5.1

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

$x = \frac{π}{2}$

Step 2.6

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

$x = π - \frac{π}{2}$

Step 2.7

Simplify $π - \frac{π}{2}$ .

Tap for more steps...

Step 2.7.1

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

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

Step 2.7.2

Combine fractions.

Tap for more steps...

Step 2.7.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 2.7.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Step 2.7.3

Simplify the numerator.

Tap for more steps...

Step 2.7.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Step 2.7.3.2

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

Step 2.8

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

Tap for more steps...

Step 2.8.1

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

$\frac{2 π}{| b |}$

Step 2.8.2

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

$\frac{2 π}{| 1 |}$

Step 2.8.3

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

$\frac{2 π}{1}$

Step 2.8.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.9

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

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

Step 3

The values which make the derivative equal to $0$ are $\frac{π}{2} + 2 π n$ .

$\frac{π}{2} + 2 π n$

Step 4

After finding the point that makes the derivative $f' (x) = 4 - 4 \sin (x)$ equal to $0$ or undefined, the interval to check where $f (x) = 4 x + 4 \cos (x)$ is increasing and where it is decreasing is $(- \infty, \frac{π}{2} + 2 π n) \cup (\frac{π}{2} + 2 π n, \infty)$ .

$(- \infty, \frac{π}{2} + 2 π n) \cup (\frac{π}{2} + 2 π n, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{π}{2} + 2 π n)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $\frac{π}{2} + 2 π n - 1$ in the expression.

$f' (\frac{π}{2} + 2 π n - 1) = 4 - 4 \sin (\frac{π}{2} + 2 π n - 1)$

Step 5.2

The final answer is $4 - 4 \sin (\frac{π}{2} + 2 π n - 1)$ .

$4 - 4 \sin (\frac{π}{2} + 2 π n - 1)$

Step 5.3

Simplify.

$4 - 4 \sin (\frac{π}{2} + 2 π n - 1)$

Step 5.4

At $x = \frac{π}{2} + 2 π n - 1$ the derivative is $4 - 4 \sin (\frac{π}{2} + 2 π n - 1)$ . Since this is negative, the function is decreasing on $(- \infty, \frac{π}{2} + 2 π n)$ .

Decreasing on $(- \infty, \frac{π}{2} + 2 π n)$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(\frac{π}{2} + 2 π n, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $\frac{π}{2} + 2 π n + 1$ in the expression.

$f' (\frac{π}{2} + 2 π n + 1) = 4 - 4 \sin (\frac{π}{2} + 2 π n + 1)$

Step 6.2

The final answer is $4 - 4 \sin (\frac{π}{2} + 2 π n + 1)$ .

$4 - 4 \sin (\frac{π}{2} + 2 π n + 1)$

Step 6.3

Simplify.

$4 - 4 \sin (\frac{π}{2} + 2 π n + 1)$

Step 6.4

At $x = \frac{π}{2} + 2 π n + 1$ the derivative is $4 - 4 \sin (\frac{π}{2} + 2 π n + 1)$ . Since this is negative, the function is decreasing on $(\frac{π}{2} + 2 π n, \infty)$ .

Decreasing on $(\frac{π}{2} + 2 π n, \infty)$ since $f' (x) < 0$

Step 7

List the intervals on which the function is increasing and decreasing.

Decreasing on: $(- \infty, \frac{π}{2} + 2 π n), (\frac{π}{2} + 2 π n, \infty)$

Step 8