Find the Derivative - d/dx y=((x+3)(x+2))/((x-3)(x-2))

Problem

d()/d(x)((x+3)*(x+2))/((x−3)*(x−2))

Solution

1. Expand the numerator and the denominator to simplify the expression before differentiating.

(x+3)*(x+2)=x2+5*x+6

(x−3)*(x−2)=x2−5*x+6

2. Rewrite the function y using the expanded forms.

y=(x2+5*x+6)/(x2−5*x+6)

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

u=x2+5*x+6⇒d(u)/d(x)=2*x+5

v=x2−5*x+6⇒d(v)/d(x)=2*x−5

4. Substitute these components into the quotient rule formula.

d(y)/d(x)=((x2−5*x+6)*(2*x+5)−(x2+5*x+6)*(2*x−5))/((x2−5*x+6)2)

5. Expand the terms in the numerator.

(x2−5*x+6)*(2*x+5)=2*x3+5*x2−10*x2−25*x+12*x+30=2*x3−5*x2−13*x+30

(x2+5*x+6)*(2*x−5)=2*x3−5*x2+10*x2−25*x+12*x−30=2*x3+5*x2−13*x−30

6. Subtract the expanded terms in the numerator and simplify.

(2*x3−5*x2−13*x+30)−(2*x3+5*x2−13*x−30)=−10*x2+60

7. Factor the resulting numerator.

−10*x2+60=−10*(x2−6)

Final Answer

d()/d(x)((x+3)*(x+2))/((x−3)*(x−2))=(−10*(x2−6))/((x2−5*x+6)2)

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