Find the Derivative - d/d@VAR f(x)=x/(x^2-9)

Problem

d()/d(x)x/(x^2−9)

Solution

1. Identify the function as a quotient of two terms, u=x and v=x^2−9 which requires the quotient rule.

2. Recall the quotient rule formula for a derivative.

d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v^2)

3. Differentiate the numerator u=x with respect to x

d(x)/d(x)=1

4. Differentiate the denominator v=x^2−9 with respect to x

d(x^2−9)/d(x)=2*x

5. Substitute these components into the quotient rule formula.

d()/d(x)x/(x^2−9)=((x^2−9)*(1)−(x)*(2*x))/((x^2−9)^2)

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

d()/d(x)x/(x^2−9)=(x^2−9−2*x^2)/((x^2−9)^2)

7. Combine the x^2 terms to reach the final simplified form.

d()/d(x)x/(x^2−9)=(−x^2−9)/((x^2−9)^2)

Final Answer

d()/d(x)x/(x^2−9)=−(x^2+9)/((x^2−9)^2)

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