Calculus Examples

Find dy/dx x^9(x+y)=y^2(4x-y)
Step 1
Differentiate both sides of the equation.
Step 2
Differentiate the left side of the equation.
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Step 2.1
Differentiate using the Product Rule which states that is where and .
Step 2.2
Differentiate.
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Step 2.2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.3
Rewrite as .
Step 2.4
Differentiate using the Power Rule which states that is where .
Step 2.5
Move to the left of .
Step 2.6
Simplify.
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Step 2.6.1
Apply the distributive property.
Step 2.6.2
Apply the distributive property.
Step 2.6.3
Apply the distributive property.
Step 2.6.4
Combine terms.
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Step 2.6.4.1
Multiply by .
Step 2.6.4.2
Multiply by by adding the exponents.
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Step 2.6.4.2.1
Move .
Step 2.6.4.2.2
Multiply by .
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Step 2.6.4.2.2.1
Raise to the power of .
Step 2.6.4.2.2.2
Use the power rule to combine exponents.
Step 2.6.4.2.3
Add and .
Step 2.6.4.3
Add and .
Step 2.6.5
Reorder terms.
Step 3
Differentiate the right side of the equation.
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Step 3.1
Differentiate using the Product Rule which states that is where and .
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
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.3
Differentiate using the Power Rule which states that is where .
Step 3.2.4
Multiply by .
Step 3.2.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
Rewrite as .
Step 3.4
Differentiate using the chain rule, which states that is where and .
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Step 3.4.1
To apply the Chain Rule, set as .
Step 3.4.2
Differentiate using the Power Rule which states that is where .
Step 3.4.3
Replace all occurrences of with .
Step 3.5
Move to the left of .
Step 3.6
Rewrite as .
Step 3.7
Simplify.
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Step 3.7.1
Apply the distributive property.
Step 3.7.2
Apply the distributive property.
Step 3.7.3
Apply the distributive property.
Step 3.7.4
Apply the distributive property.
Step 3.7.5
Combine terms.
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Step 3.7.5.1
Move to the left of .
Step 3.7.5.2
Multiply by .
Step 3.7.5.3
Multiply by .
Step 3.7.5.4
Raise to the power of .
Step 3.7.5.5
Raise to the power of .
Step 3.7.5.6
Use the power rule to combine exponents.
Step 3.7.5.7
Add and .
Step 3.7.5.8
Subtract from .
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Step 3.7.5.8.1
Reorder and .
Step 3.7.5.8.2
Subtract from .
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
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 5.2
Subtract from both sides of the equation.
Step 5.3
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
Move the negative in front of the fraction.
Step 5.5.3.2
Combine the numerators over the common denominator.
Step 5.5.3.3
Combine the numerators over the common denominator.
Step 5.5.3.4
Factor out of .
Step 5.5.3.5
Factor out of .
Step 5.5.3.6
Factor out of .
Step 5.5.3.7
Factor out of .
Step 5.5.3.8
Factor out of .
Step 5.5.3.9
Rewrite negatives.
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Step 5.5.3.9.1
Rewrite as .
Step 5.5.3.9.2
Move the negative in front of the fraction.
Step 6
Replace with .