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Calculus Examples
Step 1
Set as a function of .
Step 2
Step 2.1
Differentiate.
Step 2.1.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.2
Differentiate using the Power Rule which states that is where .
Step 2.2
The derivative of with respect to is .
Step 3
Step 3.1
Subtract from both sides of the equation.
Step 3.2
Divide each term in by and simplify.
Step 3.2.1
Divide each term in by .
Step 3.2.2
Simplify the left side.
Step 3.2.2.1
Dividing two negative values results in a positive value.
Step 3.2.2.2
Divide by .
Step 3.2.3
Simplify the right side.
Step 3.2.3.1
Divide by .
Step 3.3
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 3.4
Simplify the right side.
Step 3.4.1
The exact value of is .
Step 3.5
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 3.6
Simplify .
Step 3.6.1
To write as a fraction with a common denominator, multiply by .
Step 3.6.2
Combine fractions.
Step 3.6.2.1
Combine and .
Step 3.6.2.2
Combine the numerators over the common denominator.
Step 3.6.3
Simplify the numerator.
Step 3.6.3.1
Move to the left of .
Step 3.6.3.2
Subtract from .
Step 3.7
Find the period of .
Step 3.7.1
The period of the function can be calculated using .
Step 3.7.2
Replace with in the formula for period.
Step 3.7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.7.4
Divide by .
Step 3.8
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 4
Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
Step 4.2.1
The exact value of is .
Step 4.2.2
Add and .
Step 4.2.3
The final answer is .
Step 5
Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
Step 5.2.1
Find the common denominator.
Step 5.2.1.1
Write as a fraction with denominator .
Step 5.2.1.2
Multiply by .
Step 5.2.1.3
Multiply by .
Step 5.2.1.4
Write as a fraction with denominator .
Step 5.2.1.5
Multiply by .
Step 5.2.1.6
Multiply by .
Step 5.2.2
Combine the numerators over the common denominator.
Step 5.2.3
Simplify each term.
Step 5.2.3.1
Multiply by .
Step 5.2.3.2
To write as a fraction with a common denominator, multiply by .
Step 5.2.3.3
Combine and .
Step 5.2.3.4
Combine the numerators over the common denominator.
Step 5.2.3.5
Simplify the numerator.
Step 5.2.3.5.1
Multiply by .
Step 5.2.3.5.2
Add and .
Step 5.2.3.6
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 5.2.3.7
The exact value of is .
Step 5.2.3.8
Multiply by .
Step 5.2.4
Simplify by adding terms.
Step 5.2.4.1
Add and .
Step 5.2.4.2
Add and .
Step 5.2.5
The final answer is .
Step 6
The horizontal tangent line on function is .
Step 7