Find the Derivative - d/d@VAR f(x)=x/(x-1)

Problem

d()/d(x)x/(x−1)

Solution

1. Identify the function as a quotient of two terms, u=x and v=x−1 which requires the quotient rule.

2. Recall the quotient rule formula for a function u/v which is (vd(u)/d(x)−ud(v)/d(x))/(v2)

3. Differentiate the numerator u=x to get d(x)/d(x)=1

4. Differentiate the denominator v=x−1 to get d(x−1)/d(x)=1

5. Substitute these values into the quotient rule formula.

((x−1)*(1)−(x)*(1))/((x−1)2)

6. Simplify the numerator by distributing and combining like terms.

(x−1−x)/((x−1)2)

7. Finalize the expression by canceling x and −x

(−1)/((x−1)2)

Final Answer

d()/d(x)x/(x−1)=−1/((x−1)2)

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