Find the Derivative - d/dx y=(3x-5)/(2x+3)

Problem

d()/d(x)(3*x−5)/(2*x+3)

Solution

1. Identify the rule needed for differentiation. Since the function is a ratio of two expressions, 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=3*x−5 and v=2*x+3

3. Differentiate the individual components.

d(u)/d(x)=3

d(v)/d(x)=2

4. Substitute these values into the quotient rule formula.

d(y)/d(x)=((2*x+3)*(3)−(3*x−5)*(2))/((2*x+3)2)

5. Distribute the constants in the numerator.

d(y)/d(x)=(6*x+9−(6*x−10))/((2*x+3)2)

6. Simplify the numerator by combining like terms.

d(y)/d(x)=(6*x+9−6*x+10)/((2*x+3)2)

d(y)/d(x)=19/((2*x+3)2)

Final Answer

d()/d(x)(3*x−5)/(2*x+3)=19/((2*x+3)2)

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