Calculus Examples

Find dy/dx y=(x^2-3y)^3
Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
Step 3
Differentiate the right side of the equation.
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Step 3.1
Differentiate using the chain rule, which states that is where and .
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Step 3.1.1
To apply the Chain Rule, set as .
Step 3.1.2
Differentiate using the Power Rule which states that is where .
Step 3.1.3
Replace all occurrences of with .
Step 3.2
Differentiate.
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Step 3.2.1
By the Sum Rule, 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
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
Rewrite as .
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
Rewrite as .
Step 5.1.2
Expand using the FOIL Method.
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Step 5.1.2.1
Apply the distributive property.
Step 5.1.2.2
Apply the distributive property.
Step 5.1.2.3
Apply the distributive property.
Step 5.1.3
Simplify and combine like terms.
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Step 5.1.3.1
Simplify each term.
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Step 5.1.3.1.1
Multiply by by adding the exponents.
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Step 5.1.3.1.1.1
Use the power rule to combine exponents.
Step 5.1.3.1.1.2
Add and .
Step 5.1.3.1.2
Rewrite using the commutative property of multiplication.
Step 5.1.3.1.3
Rewrite using the commutative property of multiplication.
Step 5.1.3.1.4
Multiply by by adding the exponents.
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Step 5.1.3.1.4.1
Move .
Step 5.1.3.1.4.2
Multiply by .
Step 5.1.3.1.5
Multiply by .
Step 5.1.3.2
Subtract from .
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Step 5.1.3.2.1
Move .
Step 5.1.3.2.2
Subtract from .
Step 5.1.4
Apply the distributive property.
Step 5.1.5
Simplify.
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Step 5.1.5.1
Multiply by .
Step 5.1.5.2
Multiply by .
Step 5.1.6
Expand by multiplying each term in the first expression by each term in the second expression.
Step 5.1.7
Simplify each term.
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Step 5.1.7.1
Rewrite using the commutative property of multiplication.
Step 5.1.7.2
Multiply by by adding the exponents.
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Step 5.1.7.2.1
Move .
Step 5.1.7.2.2
Multiply by .
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Step 5.1.7.2.2.1
Raise to the power of .
Step 5.1.7.2.2.2
Use the power rule to combine exponents.
Step 5.1.7.2.3
Add and .
Step 5.1.7.3
Multiply by .
Step 5.1.7.4
Rewrite using the commutative property of multiplication.
Step 5.1.7.5
Multiply by .
Step 5.1.7.6
Multiply by by adding the exponents.
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Step 5.1.7.6.1
Move .
Step 5.1.7.6.2
Multiply by .
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Step 5.1.7.6.2.1
Raise to the power of .
Step 5.1.7.6.2.2
Use the power rule to combine exponents.
Step 5.1.7.6.3
Add and .
Step 5.1.7.7
Multiply by .
Step 5.1.7.8
Multiply by .
Step 5.1.7.9
Rewrite using the commutative property of multiplication.
Step 5.1.7.10
Multiply by .
Step 5.1.7.11
Rewrite using the commutative property of multiplication.
Step 5.1.7.12
Multiply by .
Step 5.2
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 5.3
Subtract from both sides of the equation.
Step 5.4
Move all terms not containing to the right side of the equation.
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Step 5.4.1
Subtract from both sides of the equation.
Step 5.4.2
Add to both sides of the equation.
Step 5.4.3
Subtract from both sides of the equation.
Step 5.5
Factor out of .
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Step 5.5.1
Factor out of .
Step 5.5.2
Factor out of .
Step 5.5.3
Factor out of .
Step 5.5.4
Factor out of .
Step 5.5.5
Factor out of .
Step 5.5.6
Factor out of .
Step 5.5.7
Factor out of .
Step 5.6
Divide each term in by and simplify.
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Step 5.6.1
Divide each term in by .
Step 5.6.2
Simplify the left side.
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Step 5.6.2.1
Cancel the common factor of .
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Step 5.6.2.1.1
Cancel the common factor.
Step 5.6.2.1.2
Divide by .
Step 5.6.3
Simplify the right side.
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Step 5.6.3.1
Simplify terms.
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Step 5.6.3.1.1
Simplify each term.
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Step 5.6.3.1.1.1
Move the negative in front of the fraction.
Step 5.6.3.1.1.2
Move the negative in front of the fraction.
Step 5.6.3.1.2
Combine into one fraction.
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Step 5.6.3.1.2.1
Combine the numerators over the common denominator.
Step 5.6.3.1.2.2
Combine the numerators over the common denominator.
Step 5.6.3.2
Simplify the numerator.
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Step 5.6.3.2.1
Factor out of .
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Step 5.6.3.2.1.1
Factor out of .
Step 5.6.3.2.1.2
Factor out of .
Step 5.6.3.2.1.3
Factor out of .
Step 5.6.3.2.1.4
Factor out of .
Step 5.6.3.2.1.5
Factor out of .
Step 5.6.3.2.2
Factor by grouping.
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Step 5.6.3.2.2.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Step 5.6.3.2.2.1.1
Reorder terms.
Step 5.6.3.2.2.1.2
Reorder and .
Step 5.6.3.2.2.1.3
Factor out of .
Step 5.6.3.2.2.1.4
Rewrite as plus
Step 5.6.3.2.2.1.5
Apply the distributive property.
Step 5.6.3.2.2.1.6
Move parentheses.
Step 5.6.3.2.2.2
Factor out the greatest common factor from each group.
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Step 5.6.3.2.2.2.1
Group the first two terms and the last two terms.
Step 5.6.3.2.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 5.6.3.2.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 5.6.3.2.3
Combine exponents.
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Step 5.6.3.2.3.1
Factor out of .
Step 5.6.3.2.3.2
Factor out of .
Step 5.6.3.2.3.3
Factor out of .
Step 5.6.3.2.3.4
Rewrite as .
Step 5.6.3.2.3.5
Raise to the power of .
Step 5.6.3.2.3.6
Raise to the power of .
Step 5.6.3.2.3.7
Use the power rule to combine exponents.
Step 5.6.3.2.3.8
Add and .
Step 5.6.3.2.3.9
Multiply by .
Step 5.6.3.3
Simplify with factoring out.
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Step 5.6.3.3.1
Move the negative in front of the fraction.
Step 5.6.3.3.2
Factor out of .
Step 5.6.3.3.3
Factor out of .
Step 5.6.3.3.4
Factor out of .
Step 5.6.3.3.5
Factor out of .
Step 5.6.3.3.6
Factor out of .
Step 5.6.3.3.7
Rewrite as .
Step 5.6.3.3.8
Factor out of .
Step 5.6.3.3.9
Simplify the expression.
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Step 5.6.3.3.9.1
Rewrite as .
Step 5.6.3.3.9.2
Move the negative in front of the fraction.
Step 5.6.3.3.9.3
Multiply by .
Step 5.6.3.3.9.4
Multiply by .
Step 6
Replace with .