Calculus Examples

Find the Derivative - d/dy 1/3*( square root of y(y-3))
Step 1
Combine fractions.
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Step 1.1
Combine and .
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
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
Since is constant with respect to , the derivative of with respect to is .
Step 3.6
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
Move the negative in front of the fraction.
Step 9
Combine and .
Step 10
Multiply by .
Step 11
Simplify the expression.
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Step 11.1
Multiply by .
Step 11.2
Move to the denominator using the negative exponent rule .
Step 12
To write as a fraction with a common denominator, multiply by .
Step 13
Combine and .
Step 14
Combine the numerators over the common denominator.
Step 15
Combine and .
Step 16
Factor out of .
Step 17
Cancel the common factors.
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Step 17.1
Factor out of .
Step 17.2
Cancel the common factor.
Step 17.3
Rewrite the expression.
Step 18
Simplify.
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Step 18.1
Apply the distributive property.
Step 18.2
Simplify the numerator.
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Step 18.2.1
Simplify each term.
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Step 18.2.1.1
Combine and .
Step 18.2.1.2
Move to the numerator using the negative exponent rule .
Step 18.2.1.3
Multiply by by adding the exponents.
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Step 18.2.1.3.1
Multiply by .
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Step 18.2.1.3.1.1
Raise to the power of .
Step 18.2.1.3.1.2
Use the power rule to combine exponents.
Step 18.2.1.3.2
Write as a fraction with a common denominator.
Step 18.2.1.3.3
Combine the numerators over the common denominator.
Step 18.2.1.3.4
Subtract from .
Step 18.2.1.4
Combine and .
Step 18.2.1.5
Move the negative in front of the fraction.
Step 18.2.2
To write as a fraction with a common denominator, multiply by .
Step 18.2.3
Combine and .
Step 18.2.4
Combine the numerators over the common denominator.
Step 18.2.5
Add and .
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Step 18.2.5.1
Reorder and .
Step 18.2.5.2
Add and .
Step 18.3
Combine terms.
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Step 18.3.1
Multiply by .
Step 18.3.2
Combine.
Step 18.3.3
Apply the distributive property.
Step 18.3.4
Cancel the common factor of .
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Step 18.3.4.1
Cancel the common factor.
Step 18.3.4.2
Rewrite the expression.
Step 18.3.5
Multiply by .
Step 18.3.6
Combine and .
Step 18.3.7
Multiply by .
Step 18.3.8
Factor out of .
Step 18.3.9
Cancel the common factors.
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Step 18.3.9.1
Factor out of .
Step 18.3.9.2
Cancel the common factor.
Step 18.3.9.3
Rewrite the expression.
Step 18.3.10
Move the negative in front of the fraction.
Step 18.3.11
Multiply by .
Step 18.3.12
Factor out of .
Step 18.3.13
Factor out of .
Step 18.3.14
Factor out of .
Step 18.3.15
Cancel the common factors.
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Step 18.3.15.1
Factor out of .
Step 18.3.15.2
Cancel the common factor.
Step 18.3.15.3
Rewrite the expression.
Step 18.4
Simplify the numerator.
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Step 18.4.1
To write as a fraction with a common denominator, multiply by .
Step 18.4.2
Combine the numerators over the common denominator.
Step 18.4.3
Simplify the numerator.
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Step 18.4.3.1
Multiply by by adding the exponents.
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Step 18.4.3.1.1
Use the power rule to combine exponents.
Step 18.4.3.1.2
Combine the numerators over the common denominator.
Step 18.4.3.1.3
Add and .
Step 18.4.3.1.4
Divide by .
Step 18.4.3.2
Simplify .
Step 18.5
Multiply the numerator by the reciprocal of the denominator.
Step 18.6
Multiply by .
Step 18.7
Move to the left of .