Find the Derivative - d/dx (x^2-4)/(x-2)

Problem

d()/d(x)(x2−4)/(x−2)

Solution

1. Factor the numerator of the expression using the difference of squares formula, a2−b2=(a−b)*(a+b)

x2−4=(x−2)*(x+2)

2. Simplify the rational expression by canceling the common factor (x−2) from the numerator and the denominator, noting that x≠2

((x−2)*(x+2))/(x−2)=x+2

3. Rewrite the derivative problem using the simplified expression.

d()/d(x)(x2−4)/(x−2)=(d(x)+2)/d(x)

4. Apply the sum rule for derivatives, which states that the derivative of a sum is the sum of the derivatives.

(d(x)+2)/d(x)=d(x)/d(x)+d(2)/d(x)

5. Evaluate the derivatives of the individual terms using the power rule and the constant rule.

d(x)/d(x)=1

d(2)/d(x)=0

6. Combine the results to find the final derivative.

1+0=1

Final Answer

d()/d(x)(x2−4)/(x−2)=1

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