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 .
Differentiate using the Power Rule which states that is where .
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 .
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 .
Reorder terms.
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 .
Factor using the rational roots test.
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If a polynomial function has integer coefficients, then every rational zero will have the form where is a factor of the constant and is a factor of the leading coefficient.
Find every combination of . These are the possible roots of the polynomial function.
Substitute and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.
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Substitute into the polynomial.
Simplify each term.
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Raise to the power of .
Multiply by .
Multiply by .
Add and .
Add and .
Since is a known root, divide the polynomial by to find the quotient polynomial. This polynomial can then be used to find the remaining roots.
Divide by .
Write as a set of factors.
Set equal to and solve for .
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Set the factor equal to .
Subtract from both sides of the equation.
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 .
Move the negative in front of the fraction.
Set equal to and solve for .
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Set the factor equal to .
Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
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Simplify the numerator.
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Raise to the power of .
Multiply by .
Multiply by .
Subtract from .
Rewrite as .
Rewrite as .
Rewrite as .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Move to the left of .
Multiply by .
Simplify .
Simplify the expression to solve for the portion of the .
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Simplify the numerator.
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Raise to the power of .
Multiply by .
Multiply by .
Subtract from .
Rewrite as .
Rewrite as .
Rewrite as .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Move to the left of .
Multiply by .
Simplify .
Change the to .
Split the fraction into two fractions.
Simplify the expression to solve for the portion of the .
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Simplify the numerator.
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Raise to the power of .
Multiply by .
Multiply by .
Subtract from .
Rewrite as .
Rewrite as .
Rewrite as .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Move to the left of .
Multiply by .
Simplify .
Change the to .
Split the fraction into two fractions.
Move the negative in front of the fraction.
The final answer is the combination of both solutions.
The solution is the result of and .
Solve for .
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Remove the parentheses around the expression .
Simplify .
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Simplify each term.
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Use the power rule to distribute the exponent.
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Apply the product rule to .
Apply the product rule to .
Multiply by by adding the exponents.
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Move .
Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Add and .
Raise to the power of .
One to any power is one.
Raise to the power of .
Use the power rule to distribute the exponent.
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Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Multiply by .
One to any power is one.
Raise to the power of .
Cancel the common factor of .
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Write as a fraction with denominator .
Factor out the greatest common factor .
Cancel the common factor.
Rewrite the expression.
Multiply and .
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 .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
Move the negative in front of the fraction.
Combine the numerators over the common denominator.
Add and .
Reduce the expression by cancelling the common factors.
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
Calculated values cannot contain imaginary components.
is not a valid value for x
Calculated values cannot contain imaginary components.
is not a valid value for x
Find the points where .
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