Find the Derivative - d/d@VAR f(x)=(ax+b)/(cx+d)

Problem

d()/d(x)(a*x+b)/(c*x+d)

Solution

1. Identify the function as a quotient of two linear expressions, where u=a*x+b and v=c*x+d

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

3. Differentiate the numerator u=a*x+b to get d(u)/d(x)=a

4. Differentiate the denominator v=c*x+d to get d(v)/d(x)=c

5. Substitute these derivatives into the quotient rule formula:

d()/d(x)(a*x+b)/(c*x+d)=((c*x+d)*(a)−(a*x+b)*(c))/((c*x+d)2)

6. Expand the terms in the numerator:

(a*c*x+a*d−a*c*x−b*c)/((c*x+d)2)

7. Simplify the numerator by canceling the a*c*x and −a*c*x terms.

(a*d−b*c)/((c*x+d)2)

Final Answer

d()/d(x)(a*x+b)/(c*x+d)=(a*d−b*c)/((c*x+d)2)

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