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

Problem

d()/d(x)(3*x)/(2*y)

Solution

1. Identify the expression as a quotient of two functions where y is implicitly a function of x denoted as y(x)

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

3. Differentiate the numerator 3*x with respect to x which results in 3

4. Differentiate the denominator 2*y with respect to x using the chain rule, which results in 2d(y)/d(x)

5. Substitute these values into the quotient rule formula:

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

6. Simplify the expression by performing the multiplications in the numerator and squaring the denominator:

d()/d(x)(3*x)/(2*y)=(6*y−6*xd(y)/d(x))/(4*y2)

7. Factor out a common factor of 2 from the numerator and denominator to reduce the fraction:

d()/d(x)(3*x)/(2*y)=(3*y−3*xd(y)/d(x))/(2*y2)

Final Answer

d()/d(x)(3*x)/(2*y)=(3*y−3*xd(y)/d(x))/(2*y2)

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