Calculus Examples

Find the Second Derivative f(x)=(x+1)/(x-1)
Step 1
Find the first derivative.
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Step 1.1
Differentiate using the Quotient Rule which states that is where and .
Step 1.2
Differentiate.
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Step 1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.4
Simplify the expression.
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Step 1.2.4.1
Add and .
Step 1.2.4.2
Multiply by .
Step 1.2.5
By the Sum Rule, the derivative of with respect to is .
Step 1.2.6
Differentiate using the Power Rule which states that is where .
Step 1.2.7
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.8
Simplify the expression.
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Step 1.2.8.1
Add and .
Step 1.2.8.2
Multiply by .
Step 1.3
Simplify.
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Step 1.3.1
Apply the distributive property.
Step 1.3.2
Simplify the numerator.
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Step 1.3.2.1
Combine the opposite terms in .
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Step 1.3.2.1.1
Subtract from .
Step 1.3.2.1.2
Subtract from .
Step 1.3.2.2
Multiply by .
Step 1.3.2.3
Subtract from .
Step 1.3.3
Move the negative in front of the fraction.
Step 2
Find the second derivative.
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Step 2.1
Differentiate using the Constant Multiple Rule.
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Step 2.1.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.2
Apply basic rules of exponents.
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Step 2.1.2.1
Rewrite as .
Step 2.1.2.2
Multiply the exponents in .
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Step 2.1.2.2.1
Apply the power rule and multiply exponents, .
Step 2.1.2.2.2
Multiply by .
Step 2.2
Differentiate using the chain rule, which states that is where and .
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Step 2.2.1
To apply the Chain Rule, set as .
Step 2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3
Replace all occurrences of with .
Step 2.3
Differentiate.
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Step 2.3.1
Multiply by .
Step 2.3.2
By the Sum Rule, the derivative of with respect to is .
Step 2.3.3
Differentiate using the Power Rule which states that is where .
Step 2.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.5
Simplify the expression.
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Step 2.3.5.1
Add and .
Step 2.3.5.2
Multiply by .
Step 2.4
Simplify.
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Step 2.4.1
Rewrite the expression using the negative exponent rule .
Step 2.4.2
Combine and .
Step 3
The second derivative of with respect to is .