Calculus Examples

Find dy/dx 5x^3+xy^2=5x^3y^3
Step 1
Differentiate both sides of the equation.
Step 2
Differentiate the left side of the equation.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Evaluate .
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Step 2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3
Multiply by .
Step 2.3
Evaluate .
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Step 2.3.1
Differentiate using the Product Rule which states that is where and .
Step 2.3.2
Differentiate using the chain rule, which states that is where and .
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Step 2.3.2.1
To apply the Chain Rule, set as .
Step 2.3.2.2
Differentiate using the Power Rule which states that is where .
Step 2.3.2.3
Replace all occurrences of with .
Step 2.3.3
Rewrite as .
Step 2.3.4
Differentiate using the Power Rule which states that is where .
Step 2.3.5
Move to the left of .
Step 2.3.6
Multiply by .
Step 2.4
Reorder terms.
Step 3
Differentiate the right side of the equation.
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Step 3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2
Differentiate using the Product Rule which states that is where and .
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
Move to the left of .
Step 3.5
Rewrite as .
Step 3.6
Differentiate using the Power Rule which states that is where .
Step 3.7
Move to the left of .
Step 3.8
Simplify.
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Step 3.8.1
Apply the distributive property.
Step 3.8.2
Combine terms.
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Step 3.8.2.1
Multiply by .
Step 3.8.2.2
Multiply by .
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
Subtract from both sides of the equation.
Step 5.2
Move all terms not containing to the right side of the equation.
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Step 5.2.1
Subtract from both sides of the equation.
Step 5.2.2
Subtract from both sides of the equation.
Step 5.3
Factor out of .
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Step 5.3.1
Factor out of .
Step 5.3.2
Factor out of .
Step 5.3.3
Factor out of .
Step 5.4
Divide each term in by and simplify.
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Step 5.4.1
Divide each term in by .
Step 5.4.2
Simplify the left side.
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Step 5.4.2.1
Cancel the common factor of .
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Step 5.4.2.1.1
Cancel the common factor.
Step 5.4.2.1.2
Rewrite the expression.
Step 5.4.2.2
Cancel the common factor of .
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Step 5.4.2.2.1
Cancel the common factor.
Step 5.4.2.2.2
Rewrite the expression.
Step 5.4.2.3
Cancel the common factor of .
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Step 5.4.2.3.1
Cancel the common factor.
Step 5.4.2.3.2
Divide by .
Step 5.4.3
Simplify the right side.
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Step 5.4.3.1
Simplify each term.
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Step 5.4.3.1.1
Cancel the common factor of and .
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Step 5.4.3.1.1.1
Factor out of .
Step 5.4.3.1.1.2
Cancel the common factors.
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Step 5.4.3.1.1.2.1
Factor out of .
Step 5.4.3.1.1.2.2
Cancel the common factor.
Step 5.4.3.1.1.2.3
Rewrite the expression.
Step 5.4.3.1.2
Cancel the common factor of and .
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Step 5.4.3.1.2.1
Factor out of .
Step 5.4.3.1.2.2
Cancel the common factors.
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Step 5.4.3.1.2.2.1
Cancel the common factor.
Step 5.4.3.1.2.2.2
Rewrite the expression.
Step 5.4.3.1.3
Cancel the common factor of and .
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Step 5.4.3.1.3.1
Factor out of .
Step 5.4.3.1.3.2
Cancel the common factors.
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Step 5.4.3.1.3.2.1
Factor out of .
Step 5.4.3.1.3.2.2
Cancel the common factor.
Step 5.4.3.1.3.2.3
Rewrite the expression.
Step 5.4.3.1.4
Move the negative in front of the fraction.
Step 5.4.3.1.5
Cancel the common factor of and .
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Step 5.4.3.1.5.1
Factor out of .
Step 5.4.3.1.5.2
Cancel the common factors.
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Step 5.4.3.1.5.2.1
Factor out of .
Step 5.4.3.1.5.2.2
Cancel the common factor.
Step 5.4.3.1.5.2.3
Rewrite the expression.
Step 5.4.3.1.6
Move the negative in front of the fraction.
Step 5.4.3.2
To write as a fraction with a common denominator, multiply by .
Step 5.4.3.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 5.4.3.3.1
Multiply by .
Step 5.4.3.3.2
Reorder the factors of .
Step 5.4.3.4
Combine the numerators over the common denominator.
Step 5.4.3.5
Simplify the numerator.
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Step 5.4.3.5.1
Factor out of .
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Step 5.4.3.5.1.1
Factor out of .
Step 5.4.3.5.1.2
Factor out of .
Step 5.4.3.5.1.3
Factor out of .
Step 5.4.3.5.2
Multiply by by adding the exponents.
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Step 5.4.3.5.2.1
Multiply by .
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Step 5.4.3.5.2.1.1
Raise to the power of .
Step 5.4.3.5.2.1.2
Use the power rule to combine exponents.
Step 5.4.3.5.2.2
Add and .
Step 5.4.3.5.3
Rewrite as .
Step 5.4.3.5.4
Since both terms are perfect cubes, factor using the difference of cubes formula, where and .
Step 5.4.3.5.5
Simplify.
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Step 5.4.3.5.5.1
Multiply by .
Step 5.4.3.5.5.2
One to any power is one.
Step 5.4.3.6
To write as a fraction with a common denominator, multiply by .
Step 5.4.3.7
To write as a fraction with a common denominator, multiply by .
Step 5.4.3.8
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 5.4.3.8.1
Multiply by .
Step 5.4.3.8.2
Multiply by .
Step 5.4.3.8.3
Reorder the factors of .
Step 5.4.3.8.4
Reorder the factors of .
Step 5.4.3.9
Combine the numerators over the common denominator.
Step 5.4.3.10
Simplify the numerator.
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Step 5.4.3.10.1
Multiply by by adding the exponents.
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Step 5.4.3.10.1.1
Move .
Step 5.4.3.10.1.2
Multiply by .
Step 5.4.3.10.2
Apply the distributive property.
Step 5.4.3.10.3
Multiply by .
Step 5.4.3.10.4
Expand by multiplying each term in the first expression by each term in the second expression.
Step 5.4.3.10.5
Simplify each term.
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Step 5.4.3.10.5.1
Multiply by by adding the exponents.
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Step 5.4.3.10.5.1.1
Move .
Step 5.4.3.10.5.1.2
Multiply by .
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Step 5.4.3.10.5.1.2.1
Raise to the power of .
Step 5.4.3.10.5.1.2.2
Use the power rule to combine exponents.
Step 5.4.3.10.5.1.3
Add and .
Step 5.4.3.10.5.2
Multiply by by adding the exponents.
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Step 5.4.3.10.5.2.1
Move .
Step 5.4.3.10.5.2.2
Multiply by .
Step 5.4.3.10.5.3
Multiply by .
Step 5.4.3.10.5.4
Multiply by .
Step 5.4.3.10.6
Combine the opposite terms in .
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Step 5.4.3.10.6.1
Subtract from .
Step 5.4.3.10.6.2
Add and .
Step 5.4.3.10.6.3
Subtract from .
Step 5.4.3.10.6.4
Add and .
Step 5.4.3.10.7
Multiply by by adding the exponents.
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Step 5.4.3.10.7.1
Move .
Step 5.4.3.10.7.2
Multiply by .
Step 6
Replace with .