Calculus Examples

Find dy/dx x^2(x-y)^2=x^(2-y^2)
Step 1
Differentiate both sides of the equation.
Step 2
Differentiate the left side of the equation.
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Step 2.1
Rewrite as .
Step 2.2
Expand using the FOIL Method.
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Step 2.2.1
Apply the distributive property.
Step 2.2.2
Apply the distributive property.
Step 2.2.3
Apply the distributive property.
Step 2.3
Simplify and combine like terms.
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Step 2.3.1
Simplify each term.
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Step 2.3.1.1
Multiply by .
Step 2.3.1.2
Rewrite using the commutative property of multiplication.
Step 2.3.1.3
Rewrite using the commutative property of multiplication.
Step 2.3.1.4
Multiply by by adding the exponents.
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Step 2.3.1.4.1
Move .
Step 2.3.1.4.2
Multiply by .
Step 2.3.1.5
Multiply by .
Step 2.3.1.6
Multiply by .
Step 2.3.2
Subtract from .
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Step 2.3.2.1
Move .
Step 2.3.2.2
Subtract from .
Step 2.4
Differentiate using the Product Rule which states that is where and .
Step 2.5
Differentiate.
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Step 2.5.1
By the Sum Rule, the derivative of with respect to is .
Step 2.5.2
Differentiate using the Power Rule which states that is where .
Step 2.5.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.6
Differentiate using the Product Rule which states that is where and .
Step 2.7
Rewrite as .
Step 2.8
Differentiate using the Power Rule.
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Step 2.8.1
Differentiate using the Power Rule which states that is where .
Step 2.8.2
Multiply by .
Step 2.9
Differentiate using the chain rule, which states that is where and .
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Step 2.9.1
To apply the Chain Rule, set as .
Step 2.9.2
Differentiate using the Power Rule which states that is where .
Step 2.9.3
Replace all occurrences of with .
Step 2.10
Rewrite as .
Step 2.11
Differentiate using the Power Rule which states that is where .
Step 2.12
Move to the left of .
Step 2.13
Simplify.
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Step 2.13.1
Apply the distributive property.
Step 2.13.2
Apply the distributive property.
Step 2.13.3
Apply the distributive property.
Step 2.13.4
Apply the distributive property.
Step 2.13.5
Combine terms.
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Step 2.13.5.1
Multiply by by adding the exponents.
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Step 2.13.5.1.1
Move .
Step 2.13.5.1.2
Multiply by .
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Step 2.13.5.1.2.1
Raise to the power of .
Step 2.13.5.1.2.2
Use the power rule to combine exponents.
Step 2.13.5.1.3
Add and .
Step 2.13.5.2
Move to the left of .
Step 2.13.5.3
Multiply by by adding the exponents.
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Step 2.13.5.3.1
Move .
Step 2.13.5.3.2
Multiply by .
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Step 2.13.5.3.2.1
Raise to the power of .
Step 2.13.5.3.2.2
Use the power rule to combine exponents.
Step 2.13.5.3.3
Add and .
Step 2.13.5.4
Move to the left of .
Step 2.13.5.5
Raise to the power of .
Step 2.13.5.6
Use the power rule to combine exponents.
Step 2.13.5.7
Add and .
Step 2.13.5.8
Multiply by .
Step 2.13.5.9
Raise to the power of .
Step 2.13.5.10
Raise to the power of .
Step 2.13.5.11
Use the power rule to combine exponents.
Step 2.13.5.12
Add and .
Step 2.13.5.13
Add and .
Step 2.13.5.14
Subtract from .
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Step 2.13.5.14.1
Reorder and .
Step 2.13.5.14.2
Subtract from .
Step 2.13.6
Reorder terms.
Step 3
Differentiate the right side of the equation.
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Step 3.1
Differentiate using the Generalized Power Rule which states that is where and .
Step 3.2
Differentiate.
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Step 3.2.1
Subtract from .
Step 3.2.2
Differentiate using the Power Rule which states that is where .
Step 3.2.3
Multiply by .
Step 3.2.4
By the Sum Rule, the derivative of with respect to is .
Step 3.2.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.6
Add and .
Step 3.2.7
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
Differentiate using the chain rule, which states that is where and .
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Step 3.3.1
To apply the Chain Rule, set as .
Step 3.3.2
Differentiate using the Power Rule which states that is where .
Step 3.3.3
Replace all occurrences of with .
Step 3.4
Multiply by .
Step 3.5
Rewrite as .
Step 3.6
Reorder terms.
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Solve for .
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Step 5.1
Simplify .
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Step 5.1.1
Simplify each term.
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Step 5.1.1.1
Apply the distributive property.
Step 5.1.1.2
Move to the left of .
Step 5.1.1.3
Rewrite using the commutative property of multiplication.
Step 5.1.1.4
Simplify by moving inside the logarithm.
Step 5.1.1.5
Rewrite using the commutative property of multiplication.
Step 5.1.2
Reorder factors in .
Step 5.2
Add to both sides of the equation.
Step 5.3
Move all terms not containing to the right side of the equation.
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Step 5.3.1
Subtract from both sides of the equation.
Step 5.3.2
Add to both sides of the equation.
Step 5.3.3
Subtract from both sides of the equation.
Step 5.4
Factor out of .
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Step 5.4.1
Factor out of .
Step 5.4.2
Factor out of .
Step 5.4.3
Factor out of .
Step 5.4.4
Factor out of .
Step 5.4.5
Factor out of .
Step 5.5
Divide each term in by and simplify.
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Step 5.5.1
Divide each term in by .
Step 5.5.2
Simplify the left side.
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Step 5.5.2.1
Cancel the common factor of .
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Step 5.5.2.1.1
Cancel the common factor.
Step 5.5.2.1.2
Divide by .
Step 5.5.3
Simplify the right side.
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Step 5.5.3.1
Simplify terms.
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Step 5.5.3.1.1
Simplify each term.
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Step 5.5.3.1.1.1
Move the negative in front of the fraction.
Step 5.5.3.1.1.2
Move the negative in front of the fraction.
Step 5.5.3.1.1.3
Move the negative in front of the fraction.
Step 5.5.3.1.2
Simplify terms.
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Step 5.5.3.1.2.1
Combine the numerators over the common denominator.
Step 5.5.3.1.2.2
Factor out of .
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Step 5.5.3.1.2.2.1
Factor out of .
Step 5.5.3.1.2.2.2
Factor out of .
Step 5.5.3.1.2.2.3
Factor out of .
Step 5.5.3.1.2.3
Combine the numerators over the common denominator.
Step 5.5.3.2
Simplify the numerator.
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Step 5.5.3.2.1
Apply the distributive property.
Step 5.5.3.2.2
Move to the left of .
Step 5.5.3.2.3
Rewrite using the commutative property of multiplication.
Step 5.5.3.3
Simplify terms.
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Step 5.5.3.3.1
Combine the numerators over the common denominator.
Step 5.5.3.3.2
Combine the numerators over the common denominator.
Step 5.5.3.3.3
Reorder factors in .
Step 5.5.3.3.4
Factor out of .
Step 5.5.3.3.5
Factor out of .
Step 5.5.3.3.6
Factor out of .
Step 5.5.3.3.7
Factor out of .
Step 5.5.3.3.8
Factor out of .
Step 5.5.3.3.9
Rewrite negatives.
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Step 5.5.3.3.9.1
Rewrite as .
Step 5.5.3.3.9.2
Move the negative in front of the fraction.
Step 6
Replace with .