Find the Derivative - d/dx x/(x-1)

Problem

d()/d(x)x/(x−1)

Solution

1. Identify the rule needed for differentiation. Since the expression is a quotient of two functions, u=x and v=x−1 use the quotient rule.

2. Apply the quotient rule formula, which is d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

3. Differentiate the numerator and denominator separately. The derivative of the numerator is d(x)/d(x)=1 and the derivative of the denominator is d(x−1)/d(x)=1

4. Substitute these values into the quotient rule formula.

d()/d(x)x/(x−1)=((x−1)*(1)−x(1))/((x−1)2)

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

(x−1−x)/((x−1)2)

6. 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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