Calculus Examples

Find the Derivative - d/du (u- square root of u)(u+ square root of u)
Step 1
Apply basic rules of exponents.
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Step 1.1
Use to rewrite as .
Step 1.2
Use to rewrite as .
Step 2
Differentiate using the Product 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
Differentiate using the Power Rule which states that is where .
Step 4
To write as a fraction with a common denominator, multiply by .
Step 5
Combine and .
Step 6
Combine the numerators over the common denominator.
Step 7
Simplify the numerator.
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Step 7.1
Multiply by .
Step 7.2
Subtract from .
Step 8
Combine fractions.
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Step 8.1
Move the negative in front of the fraction.
Step 8.2
Combine and .
Step 8.3
Move to the denominator using the negative exponent rule .
Step 9
By the Sum Rule, the derivative of with respect to is .
Step 10
Differentiate using the Power Rule which states that is where .
Step 11
Since is constant with respect to , the derivative of with respect to is .
Step 12
Differentiate using the Power Rule which states that is where .
Step 13
To write as a fraction with a common denominator, multiply by .
Step 14
Combine and .
Step 15
Combine the numerators over the common denominator.
Step 16
Simplify the numerator.
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Step 16.1
Multiply by .
Step 16.2
Subtract from .
Step 17
Move the negative in front of the fraction.
Step 18
Combine and .
Step 19
Move to the denominator using the negative exponent rule .
Step 20
Simplify.
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Step 20.1
Reorder terms.
Step 20.2
Simplify each term.
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Step 20.2.1
Expand using the FOIL Method.
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Step 20.2.1.1
Apply the distributive property.
Step 20.2.1.2
Apply the distributive property.
Step 20.2.1.3
Apply the distributive property.
Step 20.2.2
Simplify and combine like terms.
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Step 20.2.2.1
Simplify each term.
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Step 20.2.2.1.1
Multiply by .
Step 20.2.2.1.2
Multiply by .
Step 20.2.2.1.3
Combine and .
Step 20.2.2.1.4
Move to the numerator using the negative exponent rule .
Step 20.2.2.1.5
Multiply by by adding the exponents.
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Step 20.2.2.1.5.1
Multiply by .
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Step 20.2.2.1.5.1.1
Raise to the power of .
Step 20.2.2.1.5.1.2
Use the power rule to combine exponents.
Step 20.2.2.1.5.2
Write as a fraction with a common denominator.
Step 20.2.2.1.5.3
Combine the numerators over the common denominator.
Step 20.2.2.1.5.4
Subtract from .
Step 20.2.2.1.6
Rewrite using the commutative property of multiplication.
Step 20.2.2.1.7
Cancel the common factor of .
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Step 20.2.2.1.7.1
Move the leading negative in into the numerator.
Step 20.2.2.1.7.2
Factor out of .
Step 20.2.2.1.7.3
Cancel the common factor.
Step 20.2.2.1.7.4
Rewrite the expression.
Step 20.2.2.1.8
Move the negative in front of the fraction.
Step 20.2.2.2
To write as a fraction with a common denominator, multiply by .
Step 20.2.2.3
Combine and .
Step 20.2.2.4
Combine the numerators over the common denominator.
Step 20.2.2.5
To write as a fraction with a common denominator, multiply by .
Step 20.2.2.6
Combine and .
Step 20.2.2.7
Combine the numerators over the common denominator.
Step 20.2.2.8
Combine the numerators over the common denominator.
Step 20.2.3
Simplify the numerator.
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Step 20.2.3.1
Move to the left of .
Step 20.2.3.2
Multiply by .
Step 20.2.3.3
Add and .
Step 20.2.3.4
Rewrite in a factored form.
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Step 20.2.3.4.1
Rewrite as .
Step 20.2.3.4.2
Let . Substitute for all occurrences of .
Step 20.2.3.4.3
Factor by grouping.
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Step 20.2.3.4.3.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Step 20.2.3.4.3.1.1
Factor out of .
Step 20.2.3.4.3.1.2
Rewrite as plus
Step 20.2.3.4.3.1.3
Apply the distributive property.
Step 20.2.3.4.3.1.4
Multiply by .
Step 20.2.3.4.3.2
Factor out the greatest common factor from each group.
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Step 20.2.3.4.3.2.1
Group the first two terms and the last two terms.
Step 20.2.3.4.3.2.2
Factor out the greatest common factor (GCF) from each group.
Step 20.2.3.4.3.3
Factor the polynomial by factoring out the greatest common factor, .
Step 20.2.3.4.4
Replace all occurrences of with .
Step 20.2.4
Expand using the FOIL Method.
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Step 20.2.4.1
Apply the distributive property.
Step 20.2.4.2
Apply the distributive property.
Step 20.2.4.3
Apply the distributive property.
Step 20.2.5
Simplify and combine like terms.
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Step 20.2.5.1
Simplify each term.
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Step 20.2.5.1.1
Multiply by .
Step 20.2.5.1.2
Multiply by .
Step 20.2.5.1.3
Combine and .
Step 20.2.5.1.4
Move to the numerator using the negative exponent rule .
Step 20.2.5.1.5
Multiply by by adding the exponents.
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Step 20.2.5.1.5.1
Multiply by .
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Step 20.2.5.1.5.1.1
Raise to the power of .
Step 20.2.5.1.5.1.2
Use the power rule to combine exponents.
Step 20.2.5.1.5.2
Write as a fraction with a common denominator.
Step 20.2.5.1.5.3
Combine the numerators over the common denominator.
Step 20.2.5.1.5.4
Subtract from .
Step 20.2.5.1.6
Cancel the common factor of .
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Step 20.2.5.1.6.1
Move the leading negative in into the numerator.
Step 20.2.5.1.6.2
Factor out of .
Step 20.2.5.1.6.3
Cancel the common factor.
Step 20.2.5.1.6.4
Rewrite the expression.
Step 20.2.5.1.7
Move the negative in front of the fraction.
Step 20.2.5.2
To write as a fraction with a common denominator, multiply by .
Step 20.2.5.3
Combine and .
Step 20.2.5.4
Combine the numerators over the common denominator.
Step 20.2.6
Combine the numerators over the common denominator.
Step 20.2.7
Move to the left of .
Step 20.2.8
Subtract from .
Step 20.2.9
To write as a fraction with a common denominator, multiply by .
Step 20.2.10
Combine and .
Step 20.2.11
Combine the numerators over the common denominator.
Step 20.2.12
Simplify the numerator.
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Step 20.2.12.1
Rewrite as .
Step 20.2.12.2
Let . Substitute for all occurrences of .
Step 20.2.12.3
Factor by grouping.
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Step 20.2.12.3.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Step 20.2.12.3.1.1
Multiply by .
Step 20.2.12.3.1.2
Rewrite as plus
Step 20.2.12.3.1.3
Apply the distributive property.
Step 20.2.12.3.2
Factor out the greatest common factor from each group.
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Step 20.2.12.3.2.1
Group the first two terms and the last two terms.
Step 20.2.12.3.2.2
Factor out the greatest common factor (GCF) from each group.
Step 20.2.12.3.3
Factor the polynomial by factoring out the greatest common factor, .
Step 20.2.12.4
Replace all occurrences of with .
Step 20.3
Combine the numerators over the common denominator.
Step 20.4
Simplify each term.
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Step 20.4.1
Expand using the FOIL Method.
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Step 20.4.1.1
Apply the distributive property.
Step 20.4.1.2
Apply the distributive property.
Step 20.4.1.3
Apply the distributive property.
Step 20.4.2
Simplify and combine like terms.
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Step 20.4.2.1
Simplify each term.
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Step 20.4.2.1.1
Multiply by by adding the exponents.
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Step 20.4.2.1.1.1
Move .
Step 20.4.2.1.1.2
Use the power rule to combine exponents.
Step 20.4.2.1.1.3
Combine the numerators over the common denominator.
Step 20.4.2.1.1.4
Add and .
Step 20.4.2.1.1.5
Divide by .
Step 20.4.2.1.2
Simplify .
Step 20.4.2.1.3
Multiply by .
Step 20.4.2.1.4
Multiply by .
Step 20.4.2.1.5
Multiply by .
Step 20.4.2.2
Add and .
Step 20.4.3
Expand using the FOIL Method.
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Step 20.4.3.1
Apply the distributive property.
Step 20.4.3.2
Apply the distributive property.
Step 20.4.3.3
Apply the distributive property.
Step 20.4.4
Simplify and combine like terms.
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Step 20.4.4.1
Simplify each term.
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Step 20.4.4.1.1
Multiply by by adding the exponents.
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Step 20.4.4.1.1.1
Move .
Step 20.4.4.1.1.2
Use the power rule to combine exponents.
Step 20.4.4.1.1.3
Combine the numerators over the common denominator.
Step 20.4.4.1.1.4
Add and .
Step 20.4.4.1.1.5
Divide by .
Step 20.4.4.1.2
Simplify .
Step 20.4.4.1.3
Multiply by .
Step 20.4.4.1.4
Rewrite as .
Step 20.4.4.1.5
Multiply by .
Step 20.4.4.2
Subtract from .
Step 20.5
Combine the opposite terms in .
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Step 20.5.1
Add and .
Step 20.5.2
Add and .
Step 20.6
Add and .
Step 20.7
Subtract from .
Step 20.8
Cancel the common factor of and .
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Step 20.8.1
Factor out of .
Step 20.8.2
Factor out of .
Step 20.8.3
Factor out of .
Step 20.8.4
Cancel the common factors.
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Step 20.8.4.1
Factor out of .
Step 20.8.4.2
Cancel the common factor.
Step 20.8.4.3
Rewrite the expression.
Step 20.8.4.4
Divide by .