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

Problem

d()/d(x)(x2)/(x2−9)

Solution

1. Identify the rule needed for the derivative of a quotient. Since the expression is in the form u/v we use the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the variables for the numerator and denominator. Let u=x2 and v=x2−9

3. Differentiate the individual components.

d(x2)/d(x)=2*x

d(x2−9)/d(x)=2*x

4. Substitute these components into the quotient rule formula.

d()/d(x)(x2)/(x2−9)=((x2−9)*(2*x)−(x2)*(2*x))/((x2−9)2)

5. Expand the terms in the numerator.

d()/d(x)(x2)/(x2−9)=(2*x3−18*x−2*x3)/((x2−9)2)

6. Simplify the numerator by combining like terms.

d()/d(x)(x2)/(x2−9)=(−18*x)/((x2−9)2)

Final Answer

d()/d(x)(x2)/(x2−9)=−(18*x)/((x2−9)2)

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