Find the Derivative - d/dx 3x^2y^2

Problem

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

Solution

1. Identify the type of differentiation required. Since the expression contains both x and y and we are differentiating with respect to x we must treat y as a function of x denoted as y(x) This requires the Product Rule and the Chain Rule.

2. Apply the Product Rule, which states that d()/d(x)*[ƒ(x)*g(x)]=ƒ(x)^′*g(x)+ƒ(x)*g(x)^′ Let ƒ(x)=3*x^2 and g(x)=y^2

3. Differentiate the first part, 3*x^2 with respect to x

(d(3)*x^2)/d(x)=6*x

4. Differentiate the second part, y^2 with respect to x using the Chain Rule.

d(y^2)/d(x)=2*yd(y)/d(x)

5. Combine the results using the Product Rule formula.

d()/d(x)*3*x^2*y^2=(6*x)*(y^2)+(3*x^2)*(2*yd(y)/d(x))

6. Simplify the expression by multiplying the terms.

d()/d(x)*3*x^2*y^2=6*x*y^2+6*x^2*yd(y)/d(x)

Final Answer

d()/d(x)*3*x^2*y^2=6*x*y^2+6*x^2*yd(y)/d(x)

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