Calculus Examples

Find the Derivative - d/dx (x^2-2x-1)((x+1)/(x+3))
Step 1
Differentiate using the Product Rule which states that is where and .
Step 2
Differentiate using the Quotient Rule which states that is where and .
Step 3
Differentiate.
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Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Differentiate using the Power Rule which states that is where .
Step 3.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.4
Simplify the expression.
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Step 3.4.1
Add and .
Step 3.4.2
Multiply by .
Step 3.5
By the Sum Rule, the derivative of with respect to is .
Step 3.6
Differentiate using the Power Rule which states that is where .
Step 3.7
Since is constant with respect to , the derivative of with respect to is .
Step 3.8
Simplify the expression.
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Step 3.8.1
Add and .
Step 3.8.2
Multiply by .
Step 3.9
By the Sum Rule, the derivative of with respect to is .
Step 3.10
Differentiate using the Power Rule which states that is where .
Step 3.11
Since is constant with respect to , the derivative of with respect to is .
Step 3.12
Differentiate using the Power Rule which states that is where .
Step 3.13
Multiply by .
Step 3.14
Since is constant with respect to , the derivative of with respect to is .
Step 3.15
Add and .
Step 4
Simplify.
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Step 4.1
Apply the distributive property.
Step 4.2
Combine terms.
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Step 4.2.1
Multiply by .
Step 4.2.2
Subtract from .
Step 4.2.3
Add and .
Step 4.2.4
Subtract from .
Step 4.3
Reorder terms.
Step 4.4
Simplify each term.
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Step 4.4.1
Multiply by .
Step 4.4.2
Move to the left of .
Step 4.4.3
Multiply by .
Step 4.4.4
Factor out of .
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Step 4.4.4.1
Factor out of .
Step 4.4.4.2
Factor out of .
Step 4.4.4.3
Factor out of .
Step 4.5
To write as a fraction with a common denominator, multiply by .
Step 4.6
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 4.6.1
Multiply by .
Step 4.6.2
Raise to the power of .
Step 4.6.3
Raise to the power of .
Step 4.6.4
Use the power rule to combine exponents.
Step 4.6.5
Add and .
Step 4.7
Combine the numerators over the common denominator.
Step 4.8
Simplify the numerator.
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Step 4.8.1
Factor out of .
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Step 4.8.1.1
Factor out of .
Step 4.8.1.2
Factor out of .
Step 4.8.2
Expand using the FOIL Method.
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Step 4.8.2.1
Apply the distributive property.
Step 4.8.2.2
Apply the distributive property.
Step 4.8.2.3
Apply the distributive property.
Step 4.8.3
Combine the opposite terms in .
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Step 4.8.3.1
Reorder the factors in the terms and .
Step 4.8.3.2
Subtract from .
Step 4.8.3.3
Add and .
Step 4.8.4
Simplify each term.
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Step 4.8.4.1
Multiply by .
Step 4.8.4.2
Multiply by .
Step 4.8.5
Expand using the FOIL Method.
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Step 4.8.5.1
Apply the distributive property.
Step 4.8.5.2
Apply the distributive property.
Step 4.8.5.3
Apply the distributive property.
Step 4.8.6
Simplify each term.
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Step 4.8.6.1
Multiply by by adding the exponents.
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Step 4.8.6.1.1
Multiply by .
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Step 4.8.6.1.1.1
Raise to the power of .
Step 4.8.6.1.1.2
Use the power rule to combine exponents.
Step 4.8.6.1.2
Add and .
Step 4.8.6.2
Move to the left of .
Step 4.8.6.3
Rewrite as .
Step 4.8.6.4
Multiply by .
Step 4.8.7
Add and .
Step 4.8.8
Subtract from .
Step 4.8.9
Subtract from .
Step 4.8.10
Reorder terms.