By the Sum Rule, the derivative of with respect to is .

Evaluate .

Since is constant with respect to , the derivative of with respect to is .

The derivative of with respect to is .

Evaluate .

Since is constant with respect to , the derivative of with respect to is .

The derivative of with respect to is .

Multiply by .

Set the derivative equal to .

Divide each term in the equation by .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Replace with an equivalent expression in the numerator.

Remove parentheses.

Apply the distributive property.

Multiply by .

Rewrite in terms of sines and cosines.

Apply the distributive property.

Cancel the common factor of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

Multiply .

Combine and .

Combine and .

Move the negative in front of the fraction.

Simplify each term.

Separate fractions.

Convert from to .

Divide by .

Multiply by .

Separate fractions.

Convert from to .

Divide by .

Multiply by .

Subtract from both sides of the equation.

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

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

The exact value of is .

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from to find the solution in the fourth quadrant.

Simplify .

To write as a fraction with a common denominator, multiply by .

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Move to the left of .

Add and .

Find the period.

The period of the function can be calculated using .

Replace with in the formula for period.

Solve the equation.

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

Divide by .

The period of the function is so values will repeat every radians in both directions.

, for any integer

, for any integer

The values which make the derivative equal to are .

Split into separate intervals around the values that make the derivative or undefined.

Replace the variable with in the expression.

The final answer is .

At the derivative is . Since this is negative, the function is decreasing on .

Decreasing on since

Decreasing on since

Replace the variable with in the expression.

The final answer is .

Simplify.

At the derivative is . Since this is negative, the function is decreasing on .

Decreasing on since

Decreasing on since

Replace the variable with in the expression.

The final answer is .

Simplify.

At the derivative is . Since this is positive, the function is increasing on .

Increasing on since

Increasing on since

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

Increasing on:

Decreasing on:

Find Where Increasing/Decreasing f(x)=3sin(x)+3cos(x)