Calculus Examples

Find the Derivative - d/dy p(y)=(y^-1+y^-2)(8y^-3-9y^-4)
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
Since is constant with respect to , the derivative of with respect to is .
Step 2.3
Differentiate using the Power Rule which states that is where .
Step 2.4
Multiply by .
Step 2.5
Since is constant with respect to , the derivative of with respect to is .
Step 2.6
Differentiate using the Power Rule which states that is where .
Step 2.7
Multiply by .
Step 2.8
By the Sum Rule, the derivative of with respect to is .
Step 2.9
Differentiate using the Power Rule which states that is where .
Step 2.10
Differentiate using the Power Rule which states that is where .
Step 3
Simplify.
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Step 3.1
Reorder terms.
Step 3.2
Simplify each term.
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Step 3.2.1
Simplify each term.
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Step 3.2.1.1
Rewrite the expression using the negative exponent rule .
Step 3.2.1.2
Combine and .
Step 3.2.1.3
Move the negative in front of the fraction.
Step 3.2.1.4
Rewrite the expression using the negative exponent rule .
Step 3.2.1.5
Combine and .
Step 3.2.2
Simplify each term.
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Step 3.2.2.1
Rewrite the expression using the negative exponent rule .
Step 3.2.2.2
Rewrite the expression using the negative exponent rule .
Step 3.2.3
Expand using the FOIL Method.
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Step 3.2.3.1
Apply the distributive property.
Step 3.2.3.2
Apply the distributive property.
Step 3.2.3.3
Apply the distributive property.
Step 3.2.4
Simplify and combine like terms.
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Step 3.2.4.1
Simplify each term.
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Step 3.2.4.1.1
Multiply .
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Step 3.2.4.1.1.1
Multiply by .
Step 3.2.4.1.1.2
Multiply by by adding the exponents.
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Step 3.2.4.1.1.2.1
Multiply by .
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Step 3.2.4.1.1.2.1.1
Raise to the power of .
Step 3.2.4.1.1.2.1.2
Use the power rule to combine exponents.
Step 3.2.4.1.1.2.2
Add and .
Step 3.2.4.1.2
Multiply .
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Step 3.2.4.1.2.1
Multiply by .
Step 3.2.4.1.2.2
Multiply by by adding the exponents.
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Step 3.2.4.1.2.2.1
Use the power rule to combine exponents.
Step 3.2.4.1.2.2.2
Add and .
Step 3.2.4.1.3
Combine.
Step 3.2.4.1.4
Multiply by by adding the exponents.
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Step 3.2.4.1.4.1
Multiply by .
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Step 3.2.4.1.4.1.1
Raise to the power of .
Step 3.2.4.1.4.1.2
Use the power rule to combine exponents.
Step 3.2.4.1.4.2
Add and .
Step 3.2.4.1.5
Multiply by .
Step 3.2.4.1.6
Combine.
Step 3.2.4.1.7
Multiply by by adding the exponents.
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Step 3.2.4.1.7.1
Use the power rule to combine exponents.
Step 3.2.4.1.7.2
Add and .
Step 3.2.4.1.8
Multiply by .
Step 3.2.4.2
Combine the numerators over the common denominator.
Step 3.2.4.3
Add and .
Step 3.2.5
Simplify each term.
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Step 3.2.5.1
Rewrite the expression using the negative exponent rule .
Step 3.2.5.2
Rewrite the expression using the negative exponent rule .
Step 3.2.5.3
Combine and .
Step 3.2.5.4
Move the negative in front of the fraction.
Step 3.2.6
Simplify each term.
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Step 3.2.6.1
Rewrite the expression using the negative exponent rule .
Step 3.2.6.2
Combine and .
Step 3.2.6.3
Rewrite the expression using the negative exponent rule .
Step 3.2.6.4
Combine and .
Step 3.2.6.5
Move the negative in front of the fraction.
Step 3.2.7
Expand using the FOIL Method.
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Step 3.2.7.1
Apply the distributive property.
Step 3.2.7.2
Apply the distributive property.
Step 3.2.7.3
Apply the distributive property.
Step 3.2.8
Simplify and combine like terms.
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Step 3.2.8.1
Simplify each term.
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Step 3.2.8.1.1
Multiply .
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Step 3.2.8.1.1.1
Multiply by .
Step 3.2.8.1.1.2
Multiply by by adding the exponents.
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Step 3.2.8.1.1.2.1
Use the power rule to combine exponents.
Step 3.2.8.1.1.2.2
Add and .
Step 3.2.8.1.2
Multiply .
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Step 3.2.8.1.2.1
Multiply by .
Step 3.2.8.1.2.2
Multiply by .
Step 3.2.8.1.2.3
Multiply by .
Step 3.2.8.1.2.4
Multiply by by adding the exponents.
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Step 3.2.8.1.2.4.1
Use the power rule to combine exponents.
Step 3.2.8.1.2.4.2
Add and .
Step 3.2.8.1.3
Multiply .
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Step 3.2.8.1.3.1
Multiply by .
Step 3.2.8.1.3.2
Multiply by .
Step 3.2.8.1.3.3
Multiply by by adding the exponents.
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Step 3.2.8.1.3.3.1
Use the power rule to combine exponents.
Step 3.2.8.1.3.3.2
Add and .
Step 3.2.8.1.4
Multiply .
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Step 3.2.8.1.4.1
Multiply by .
Step 3.2.8.1.4.2
Multiply by .
Step 3.2.8.1.4.3
Multiply by .
Step 3.2.8.1.4.4
Multiply by .
Step 3.2.8.1.4.5
Multiply by by adding the exponents.
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Step 3.2.8.1.4.5.1
Use the power rule to combine exponents.
Step 3.2.8.1.4.5.2
Add and .
Step 3.2.8.2
Combine the numerators over the common denominator.
Step 3.2.8.3
Subtract from .
Step 3.2.9
Move the negative in front of the fraction.
Step 3.3
Combine the numerators over the common denominator.
Step 3.4
Subtract from .
Step 3.5
Subtract from .
Step 3.6
Add and .
Step 3.7
Move the negative in front of the fraction.