Calculus Examples

Find the Derivative - d/dx y=(x/( square root of x-11))^3
Step 1
Use to rewrite as .
Step 2
Differentiate using the chain rule, which states that is where and .
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Step 2.1
To apply the Chain Rule, set as .
Step 2.2
Differentiate using the Power Rule which states that is where .
Step 2.3
Replace all occurrences of with .
Step 3
Differentiate using the Quotient Rule which states that is where and .
Step 4
Multiply the exponents in .
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Step 4.1
Apply the power rule and multiply exponents, .
Step 4.2
Cancel the common factor of .
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Step 4.2.1
Cancel the common factor.
Step 4.2.2
Rewrite the expression.
Step 5
Simplify.
Step 6
Differentiate using the Power Rule.
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Step 6.1
Differentiate using the Power Rule which states that is where .
Step 6.2
Multiply by .
Step 7
Differentiate using the chain rule, which states that is where and .
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Step 7.1
To apply the Chain Rule, set as .
Step 7.2
Differentiate using the Power Rule which states that is where .
Step 7.3
Replace all occurrences of with .
Step 8
To write as a fraction with a common denominator, multiply by .
Step 9
Combine and .
Step 10
Combine the numerators over the common denominator.
Step 11
Simplify the numerator.
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Step 11.1
Multiply by .
Step 11.2
Subtract from .
Step 12
Combine fractions.
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Step 12.1
Move the negative in front of the fraction.
Step 12.2
Combine and .
Step 12.3
Move to the denominator using the negative exponent rule .
Step 12.4
Combine and .
Step 13
By the Sum Rule, the derivative of with respect to is .
Step 14
Differentiate using the Power Rule which states that is where .
Step 15
Since is constant with respect to , the derivative of with respect to is .
Step 16
Simplify the expression.
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Step 16.1
Add and .
Step 16.2
Multiply by .
Step 17
To write as a fraction with a common denominator, multiply by .
Step 18
Combine and .
Step 19
Combine the numerators over the common denominator.
Step 20
Multiply by by adding the exponents.
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Step 20.1
Move .
Step 20.2
Use the power rule to combine exponents.
Step 20.3
Combine the numerators over the common denominator.
Step 20.4
Add and .
Step 20.5
Divide by .
Step 21
Simplify .
Step 22
Move to the left of .
Step 23
Rewrite as a product.
Step 24
Multiply by .
Step 25
Raise to the power of .
Step 26
Use the power rule to combine exponents.
Step 27
Simplify the expression.
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Step 27.1
Write as a fraction with a common denominator.
Step 27.2
Combine the numerators over the common denominator.
Step 27.3
Add and .
Step 28
Combine and .
Step 29
Move to the left of .
Step 30
Simplify.
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Step 30.1
Apply the product rule to .
Step 30.2
Apply the distributive property.
Step 30.3
Apply the distributive property.
Step 30.4
Combine terms.
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Step 30.4.1
Multiply by .
Step 30.4.2
Multiply by .
Step 30.4.3
Multiply by .
Step 30.4.4
Multiply by .
Step 30.4.5
Subtract from .
Step 30.4.6
Multiply the exponents in .
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Step 30.4.6.1
Apply the power rule and multiply exponents, .
Step 30.4.6.2
Cancel the common factor of .
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Step 30.4.6.2.1
Cancel the common factor.
Step 30.4.6.2.2
Rewrite the expression.
Step 30.4.7
Simplify.
Step 30.4.8
Multiply by .
Step 30.4.9
Raise to the power of .
Step 30.4.10
Use the power rule to combine exponents.
Step 30.4.11
Write as a fraction with a common denominator.
Step 30.4.12
Combine the numerators over the common denominator.
Step 30.4.13
Add and .
Step 30.5
Reorder terms.
Step 30.6
Factor out of .
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Step 30.6.1
Factor out of .
Step 30.6.2
Factor out of .
Step 30.6.3
Factor out of .
Step 30.7
Move to the left of .