Find the Derivative - d/dx 2xy^3

Problem

d()/d(x)*2*x*y3

Solution

1. Identify the variables involved, noting that y is typically treated as a function of x (implicit differentiation) or as a constant if the differentiation is partial. Assuming standard implicit differentiation, we use the product rule.

2. Apply the product rule which states d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x) where u=2*x and v=y3

3. Differentiate the first part: (d(2)*x)/d(x)=2

4. Differentiate the second part using the chain rule: d(y3)/d(x)=3*y2d(y)/d(x)

5. Combine the results using the product rule formula.

6. Simplify the expression by multiplying the terms.

Final Answer

d()/d(x)*2*x*y3=2*y3+6*x*y2d(y)/d(x)

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