# Calculus Examples

Step 1

Differentiate both sides of the equation.

Step 2

The derivative of with respect to is .

Step 3

Step 3.1

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

Step 3.2

Evaluate .

Step 3.2.1

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

Step 3.2.2

Differentiate using the Power Rule which states that is where .

Step 3.2.3

Multiply by .

Step 3.3

Evaluate .

Step 3.3.1

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

Step 3.3.2

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

Step 3.3.2.1

To apply the Chain Rule, set as .

Step 3.3.2.2

The derivative of with respect to is .

Step 3.3.2.3

Replace all occurrences of with .

Step 3.3.3

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

Step 3.3.4

Differentiate using the Power Rule which states that is where .

Step 3.3.5

Multiply by .

Step 3.3.6

Multiply by .

Step 3.3.7

Multiply by .

Step 4

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

Step 5

Replace with .

Step 6

Step 6.1

Subtract from both sides of the equation.

Step 6.2

Divide each term in by and simplify.

Step 6.2.1

Divide each term in by .

Step 6.2.2

Simplify the left side.

Step 6.2.2.1

Cancel the common factor of .

Step 6.2.2.1.1

Cancel the common factor.

Step 6.2.2.1.2

Divide by .

Step 6.2.3

Simplify the right side.

Step 6.2.3.1

Cancel the common factor of and .

Step 6.2.3.1.1

Factor out of .

Step 6.2.3.1.2

Cancel the common factors.

Step 6.2.3.1.2.1

Factor out of .

Step 6.2.3.1.2.2

Cancel the common factor.

Step 6.2.3.1.2.3

Rewrite the expression.

Step 6.2.3.2

Move the negative in front of the fraction.

Step 6.3

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

Step 6.4

Simplify the right side.

Step 6.4.1

The exact value of is .

Step 6.5

Divide each term in by and simplify.

Step 6.5.1

Divide each term in by .

Step 6.5.2

Simplify the left side.

Step 6.5.2.1

Cancel the common factor of .

Step 6.5.2.1.1

Cancel the common factor.

Step 6.5.2.1.2

Divide by .

Step 6.5.3

Simplify the right side.

Step 6.5.3.1

Multiply the numerator by the reciprocal of the denominator.

Step 6.5.3.2

Multiply .

Step 6.5.3.2.1

Multiply by .

Step 6.5.3.2.2

Multiply by .

Step 6.6

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.

Step 6.7

Simplify the expression to find the second solution.

Step 6.7.1

Subtract from .

Step 6.7.2

The resulting angle of is positive, less than , and coterminal with .

Step 6.7.3

Divide each term in by and simplify.

Step 6.7.3.1

Divide each term in by .

Step 6.7.3.2

Simplify the left side.

Step 6.7.3.2.1

Cancel the common factor of .

Step 6.7.3.2.1.1

Cancel the common factor.

Step 6.7.3.2.1.2

Divide by .

Step 6.7.3.3

Simplify the right side.

Step 6.7.3.3.1

Multiply the numerator by the reciprocal of the denominator.

Step 6.7.3.3.2

Multiply .

Step 6.7.3.3.2.1

Multiply by .

Step 6.7.3.3.2.2

Multiply by .

Step 6.8

Find the period of .

Step 6.8.1

The period of the function can be calculated using .

Step 6.8.2

Replace with in the formula for period.

Step 6.8.3

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

Step 6.9

Add to every negative angle to get positive angles.

Step 6.9.1

Add to to find the positive angle.

Step 6.9.2

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

Step 6.9.3

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

Step 6.9.3.1

Multiply by .

Step 6.9.3.2

Multiply by .

Step 6.9.4

Combine the numerators over the common denominator.

Step 6.9.5

Simplify the numerator.

Step 6.9.5.1

Multiply by .

Step 6.9.5.2

Subtract from .

Step 6.9.6

List the new angles.

Step 6.10

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

, for any integer

, for any integer

Step 7

Step 7.1

Simplify each term.

Step 7.1.1

Apply the distributive property.

Step 7.1.2

Cancel the common factor of .

Step 7.1.2.1

Factor out of .

Step 7.1.2.2

Cancel the common factor.

Step 7.1.2.3

Rewrite the expression.

Step 7.1.3

Cancel the common factor of .

Step 7.1.3.1

Factor out of .

Step 7.1.3.2

Cancel the common factor.

Step 7.1.3.3

Rewrite the expression.

Step 7.1.4

Multiply by .

Step 7.1.5

Apply the distributive property.

Step 7.1.6

Cancel the common factor of .

Step 7.1.6.1

Factor out of .

Step 7.1.6.2

Cancel the common factor.

Step 7.1.6.3

Rewrite the expression.

Step 7.1.7

Cancel the common factor of .

Step 7.1.7.1

Cancel the common factor.

Step 7.1.7.2

Rewrite the expression.

Step 7.2

Simplify with commuting.

Step 7.2.1

Reorder and .

Step 7.2.2

Reorder and .

Step 8

Step 8.1

Simplify each term.

Step 8.1.1

Apply the distributive property.

Step 8.1.2

Cancel the common factor of .

Step 8.1.2.1

Factor out of .

Step 8.1.2.2

Cancel the common factor.

Step 8.1.2.3

Rewrite the expression.

Step 8.1.3

Cancel the common factor of .

Step 8.1.3.1

Factor out of .

Step 8.1.3.2

Cancel the common factor.

Step 8.1.3.3

Rewrite the expression.

Step 8.1.4

Multiply by .

Step 8.1.5

Apply the distributive property.

Step 8.1.6

Cancel the common factor of .

Step 8.1.6.1

Factor out of .

Step 8.1.6.2

Cancel the common factor.

Step 8.1.6.3

Rewrite the expression.

Step 8.1.7

Cancel the common factor of .

Step 8.1.7.1

Cancel the common factor.

Step 8.1.7.2

Rewrite the expression.

Step 8.2

Simplify with commuting.

Step 8.2.1

Reorder and .

Step 8.2.2

Reorder and .

Step 9

Find the points where .

, for any integer

, for any integer

Step 10