Calculus Examples

Find the Derivative - d/dy (1/(y^2)-9/(y^4))(y+3y^3)
Step 1
Differentiate using the Product Rule which states that is where and .
Step 2
Differentiate.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Differentiate using the Power Rule which states that is where .
Step 2.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.4
Differentiate using the Power Rule which states that is where .
Step 2.5
Multiply by .
Step 2.6
By the Sum Rule, the derivative of with respect to is .
Step 2.7
Apply basic rules of exponents.
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Step 2.7.1
Rewrite as .
Step 2.7.2
Multiply the exponents in .
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Step 2.7.2.1
Apply the power rule and multiply exponents, .
Step 2.7.2.2
Multiply by .
Step 2.8
Differentiate using the Power Rule which states that is where .
Step 2.9
Since is constant with respect to , the derivative of with respect to is .
Step 2.10
Apply basic rules of exponents.
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Step 2.10.1
Rewrite as .
Step 2.10.2
Multiply the exponents in .
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Step 2.10.2.1
Apply the power rule and multiply exponents, .
Step 2.10.2.2
Multiply by .
Step 2.11
Differentiate using the Power Rule which states that is where .
Step 2.12
Multiply by .
Step 3
Simplify.
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Step 3.1
Rewrite the expression using the negative exponent rule .
Step 3.2
Rewrite the expression using the negative exponent rule .
Step 3.3
Combine terms.
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Step 3.3.1
Combine and .
Step 3.3.2
Move the negative in front of the fraction.
Step 3.3.3
Combine and .
Step 3.4
Reorder terms.
Step 3.5
Simplify each term.
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Step 3.5.1
Expand using the FOIL Method.
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Step 3.5.1.1
Apply the distributive property.
Step 3.5.1.2
Apply the distributive property.
Step 3.5.1.3
Apply the distributive property.
Step 3.5.2
Simplify and combine like terms.
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Step 3.5.2.1
Simplify each term.
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Step 3.5.2.1.1
Multiply by .
Step 3.5.2.1.2
Multiply by .
Step 3.5.2.1.3
Cancel the common factor of .
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Step 3.5.2.1.3.1
Factor out of .
Step 3.5.2.1.3.2
Cancel the common factor.
Step 3.5.2.1.3.3
Rewrite the expression.
Step 3.5.2.1.4
Cancel the common factor of .
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Step 3.5.2.1.4.1
Move the leading negative in into the numerator.
Step 3.5.2.1.4.2
Factor out of .
Step 3.5.2.1.4.3
Factor out of .
Step 3.5.2.1.4.4
Cancel the common factor.
Step 3.5.2.1.4.5
Rewrite the expression.
Step 3.5.2.1.5
Combine and .
Step 3.5.2.1.6
Multiply by .
Step 3.5.2.1.7
Move the negative in front of the fraction.
Step 3.5.2.2
Combine the numerators over the common denominator.
Step 3.5.2.3
Subtract from .
Step 3.5.3
Move the negative in front of the fraction.
Step 3.5.4
Expand using the FOIL Method.
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Step 3.5.4.1
Apply the distributive property.
Step 3.5.4.2
Apply the distributive property.
Step 3.5.4.3
Apply the distributive property.
Step 3.5.5
Simplify and combine like terms.
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Step 3.5.5.1
Simplify each term.
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Step 3.5.5.1.1
Cancel the common factor of .
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Step 3.5.5.1.1.1
Move the leading negative in into the numerator.
Step 3.5.5.1.1.2
Factor out of .
Step 3.5.5.1.1.3
Cancel the common factor.
Step 3.5.5.1.1.4
Rewrite the expression.
Step 3.5.5.1.2
Move the negative in front of the fraction.
Step 3.5.5.1.3
Cancel the common factor of .
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Step 3.5.5.1.3.1
Move the leading negative in into the numerator.
Step 3.5.5.1.3.2
Factor out of .
Step 3.5.5.1.3.3
Cancel the common factor.
Step 3.5.5.1.3.4
Rewrite the expression.
Step 3.5.5.1.4
Multiply by .
Step 3.5.5.1.5
Cancel the common factor of .
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Step 3.5.5.1.5.1
Factor out of .
Step 3.5.5.1.5.2
Cancel the common factor.
Step 3.5.5.1.5.3
Rewrite the expression.
Step 3.5.5.1.6
Rewrite using the commutative property of multiplication.
Step 3.5.5.1.7
Multiply .
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Step 3.5.5.1.7.1
Combine and .
Step 3.5.5.1.7.2
Multiply by .
Step 3.5.5.1.8
Cancel the common factor of .
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Step 3.5.5.1.8.1
Factor out of .
Step 3.5.5.1.8.2
Cancel the common factor.
Step 3.5.5.1.8.3
Rewrite the expression.
Step 3.5.5.2
Combine the numerators over the common denominator.
Step 3.5.5.3
Add and .
Step 3.6
Combine the numerators over the common denominator.
Step 3.7
Add and .
Step 3.8
Add and .
Step 3.9
Subtract from .