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

Problem

(d(x2)*e(3*x))/d(x)

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x2 and e(3*x) apply the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Differentiate the first part of the product. The derivative of x2 with respect to x is 2*x

3. Differentiate the second part of the product using the chain rule. The derivative of e(3*x) is e(3*x)⋅(d(3)*x)/d(x) which equals 3*e(3*x)

4. Substitute these components into the product rule formula.

x2⋅3*e(3*x)+e(3*x)⋅2*x

5. Simplify the expression by factoring out the common terms x and e(3*x)

x*e(3*x)*(3*x+2)

Final Answer

(d(x2)*e(3*x))/d(x)=3*x2*e(3*x)+2*x*e(3*x)

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