Calculus Examples

Find the Derivative - d/dx y = square root of x-1/( square root of x)
Step 1
By the Sum Rule, the derivative of with respect to is .
Step 2
Evaluate .
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Step 2.1
Use to rewrite as .
Step 2.2
Differentiate using the Power Rule which states that is where .
Step 2.3
To write as a fraction with a common denominator, multiply by .
Step 2.4
Combine and .
Step 2.5
Combine the numerators over the common denominator.
Step 2.6
Simplify the numerator.
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Step 2.6.1
Multiply by .
Step 2.6.2
Subtract from .
Step 2.7
Move the negative in front of the fraction.
Step 3
Evaluate .
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Step 3.1
Use to rewrite as .
Step 3.2
Differentiate using the Product Rule which states that is where and .
Step 3.3
Rewrite as .
Step 3.4
Differentiate using the chain rule, which states that is where and .
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Step 3.4.1
To apply the Chain Rule, set as .
Step 3.4.2
Differentiate using the Power Rule which states that is where .
Step 3.4.3
Replace all occurrences of with .
Step 3.5
Differentiate using the Power Rule which states that is where .
Step 3.6
Since is constant with respect to , the derivative of with respect to is .
Step 3.7
Multiply the exponents in .
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Step 3.7.1
Apply the power rule and multiply exponents, .
Step 3.7.2
Cancel the common factor of .
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Step 3.7.2.1
Factor out of .
Step 3.7.2.2
Cancel the common factor.
Step 3.7.2.3
Rewrite the expression.
Step 3.8
To write as a fraction with a common denominator, multiply by .
Step 3.9
Combine and .
Step 3.10
Combine the numerators over the common denominator.
Step 3.11
Simplify the numerator.
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Step 3.11.1
Multiply by .
Step 3.11.2
Subtract from .
Step 3.12
Move the negative in front of the fraction.
Step 3.13
Combine and .
Step 3.14
Combine and .
Step 3.15
Multiply by by adding the exponents.
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Step 3.15.1
Use the power rule to combine exponents.
Step 3.15.2
To write as a fraction with a common denominator, multiply by .
Step 3.15.3
Combine and .
Step 3.15.4
Combine the numerators over the common denominator.
Step 3.15.5
Simplify the numerator.
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Step 3.15.5.1
Multiply by .
Step 3.15.5.2
Subtract from .
Step 3.15.6
Move the negative in front of the fraction.
Step 3.16
Move to the denominator using the negative exponent rule .
Step 3.17
Multiply by .
Step 3.18
Multiply by .
Step 3.19
Multiply by .
Step 3.20
Add and .
Step 4
Simplify.
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Step 4.1
Rewrite the expression using the negative exponent rule .
Step 4.2
Multiply by .