Find the Derivative - d/dx x/(x-2)

Problem

d()/d(x)x/(x−2)

Solution

1. Identify the rule needed for the derivative. Since the expression is a fraction of two functions, apply the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator and denominator to variables. Let u=x and v=x−2

3. Differentiate the individual components. The derivative of the numerator is d(x)/d(x)=1 and the derivative of the denominator is d(x−2)/d(x)=1

4. Substitute these values into the quotient rule formula.

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

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

(x−2−x)/((x−2)2)

6. Finalize the expression by canceling the x terms in the numerator.

(−2)/((x−2)2)

Final Answer

d()/d(x)x/(x−2)=−2/((x−2)2)

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