Find the Derivative Using Quotient Rule - d/dx x/(x^2-9)

Problem

d()/d(x)x/(x2−9)

Solution

1. Identify the numerator u=x and the denominator v=x2−9 to apply the quotient rule.

2. Differentiate both parts to find d(u)/d(x)=1 and d(v)/d(x)=2*x

3. Apply the quotient rule formula d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

4. Substitute the expressions into the formula:

((x2−9)*(1)−(x)*(2*x))/((x2−9)2)

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

(x2−9−2*x2)/((x2−9)2)

6. Combine the x2 terms to reach the final form:

(−x2−9)/((x2−9)2)

Final Answer

d()/d(x)x/(x2−9)=(−x2−9)/((x2−9)2)

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