# Calculus Examples

Differentiate both sides of the equation.

The derivative of with respect to is .

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 .

Differentiate using the Power Rule which states that is where .

Multiply by to get .

Evaluate .

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

Differentiate using the chain rule, which states that is where and .

To apply the Chain Rule, set as .

The derivative of with respect to is .

Replace all occurrences of with .

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

Differentiate using the Power Rule which states that is where .

Multiply by to get .

Multiply by to get .

Multiply by to get .

Reform the equation by setting the left side equal to the right side.

Replace with .

Rewrite the equation as .

Since does not contain the variable to solve for, move it to the right side of the equation by subtracting from both sides.

Divide each term by and simplify.

Divide each term in by .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by to get .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Simplify the expression to find the first solution.

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

The exact value of is .

Divide each term by and simplify.

Divide each term in by .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by to get .

Simplify the right side of the equation.

Multiply by to get .

Simplify .

Multiply and to get .

Multiply by to get .

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from , to find a reference angle. Next, add this reference angle to to find the solution in the third quadrant.

Simplify the expression to find the second solution.

Simplify the right side.

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 to get .

Combine the numerators over the common denominator.

Simplify each term.

Simplify the numerator.

Factor out of .

Multiply by to get .

Add and to get .

Move to the left of the expression .

Multiply by to get .

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 to get .

Combine the numerators over the common denominator.

Simplify the numerator.

Move to the left of the expression .

Multiply by to get .

Add and to get .

Divide each term by and simplify.

Divide each term in by .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by to get .

Simplify the right side of the equation.

Multiply by to get .

Simplify .

Multiply and to get .

Multiply by to get .

Find the period.

The period of the function can be calculated using .

Replace with in the formula for period.

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

Add to every negative angle to get positive angles.

Add to to find the positive angle.

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 to get .

Combine the numerators over the common denominator.

Simplify the numerator.

Factor out of .

Multiply by to get .

Subtract from to get .

Simplify the expression.

Move to the left of the expression .

Multiply by to get .

List the new angles.

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

Find the points where .