Calculus Examples

Find Where dy/dx is Equal to Zero
Differentiate both sides of the equation.
The derivative of with respect to is .
Differentiate the right side of the equation.
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By the Sum Rule, the derivative of with respect to is .
Evaluate .
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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 .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the chain rule, which states that is where and .
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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 .
Set then solve for in terms of .
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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.
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Divide each term in by .
Reduce the expression by cancelling the common factors.
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Cancel the common factor.
Divide by to get .
Reduce the expression by cancelling the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify the expression to find the first solution.
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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.
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Divide each term in by .
Reduce the expression by cancelling the common factors.
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Cancel the common factor.
Divide by to get .
Simplify the right side of the equation.
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Multiply by to get .
Simplify .
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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.
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Simplify the right side.
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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 .
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Combine.
Multiply by to get .
Combine the numerators over the common denominator.
Simplify each term.
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Simplify the numerator.
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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 .
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Combine.
Multiply by to get .
Combine the numerators over the common denominator.
Simplify the numerator.
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Move to the left of the expression .
Multiply by to get .
Add and to get .
Divide each term by and simplify.
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Divide each term in by .
Reduce the expression by cancelling the common factors.
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Cancel the common factor.
Divide by to get .
Simplify the right side of the equation.
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Multiply by to get .
Simplify .
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Multiply and to get .
Multiply by to get .
Find the period.
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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.
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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 .
Tap for more steps...
Combine.
Multiply by to get .
Combine the numerators over the common denominator.
Simplify the numerator.
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Factor out of .
Multiply by to get .
Subtract from to get .
Simplify the expression.
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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 .
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