Find dy/dx y=(4x)/(x-3)

Problem

d()/d(x)(4*x)/(x−3)

Solution

1. Identify the function as a quotient of two terms, where u=4*x and v=x−3

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

3. Differentiate the numerator and denominator separately to find d(u)/d(x)=4 and d(v)/d(x)=1

4. Substitute these values into the quotient rule formula:

d(y)/d(x)=((x−3)*(4)−(4*x)*(1))/((x−3)2)

5. Distribute the constant in the numerator:

d(y)/d(x)=(4*x−12−4*x)/((x−3)2)

6. Simplify the numerator by combining like terms:

d(y)/d(x)=(−12)/((x−3)2)

Final Answer

d()/d(x)(4*x)/(x−3)=−12/((x−3)2)

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