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

Problem

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

Solution

1. Identify the variables involved, noting that y is implicitly a function of x which requires the use of the quotient rule and the chain rule.

2. Apply the quotient rule formula, which states that d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2) where u=2*x and v=3*y

3. Differentiate the numerator with respect to x which gives (d(2)*x)/d(x)=2

4. Differentiate the denominator with respect to x using the chain rule, which gives (d(3)*y)/d(x)=3d(y)/d(x)

5. Substitute these derivatives back into the quotient rule formula.

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

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

d()/d(x)(2*x)/(3*y)=(6*y−6*xd(y)/d(x))/(9*y2)

7. Factor out the common factor of 3 from the numerator and denominator to reduce the fraction.

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

Final Answer

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

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