Find the Derivative - d/dy (y-3)/(y^2-3y+9)

Problem

d()/d(y)(y−3)/(y2−3*y+9)

Solution

1. Identify the rule needed for differentiation. Since the expression is a fraction of two functions, apply the quotient rule: d()/d(y)u/v=(vd(u)/d(y)−ud(v)/d(y))/(v2)

2. Assign the numerator and denominator functions. Let u=y−3 and v=y2−3*y+9

3. Differentiate the individual components.

d(y−3)/d(y)=1

d(y2−3*y+9)/d(y)=2*y−3

4. Substitute these values into the quotient rule formula.

((y2−3*y+9)*(1)−(y−3)*(2*y−3))/((y2−3*y+9)2)

5. Expand the terms in the numerator.

(y−3)*(2*y−3)=2*y2−3*y−6*y+9

(y−3)*(2*y−3)=2*y2−9*y+9

6. Subtract the expanded terms from the first part of the numerator.

y2−3*y+9−(2*y2−9*y+9)

y2−3*y+9−2*y2+9*y−9

7. Combine like terms to simplify the numerator.

−y2+6*y

Final Answer

d()/d(y)(y−3)/(y2−3*y+9)=(−y2+6*y)/((y2−3*y+9)2)

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