Find dy/dx y=(3x-9)/(2x+8)

Problem

y=(3*x−9)/(2*x+8)

Solution

1. Identify the function as a quotient of two linear expressions, u=3*x−9 and v=2*x+8

2. Apply the quotient rule for differentiation, which states that d(y)/d(x)=(vd(u)/d(x)−ud(v)/d(x))/(v2)

3. Differentiate the numerator u and the denominator v with respect to x

d(3*x−9)/d(x)=3

d(2*x+8)/d(x)=2

4. Substitute these derivatives and the original expressions into the quotient rule formula.

d(y)/d(x)=((2*x+8)*(3)−(3*x−9)*(2))/((2*x+8)2)

5. Expand the terms in the numerator.

d(y)/d(x)=(6*x+24−(6*x−18))/((2*x+8)2)

6. Simplify the numerator by combining like terms.

d(y)/d(x)=(6*x+24−6*x+18)/((2*x+8)2)

d(y)/d(x)=42/((2*x+8)2)

Final Answer

d(y)/d(x)=42/((2*x+8)2)

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