Find the Derivative - d/dx y=5x^2e^(3x)

Problem

d()/d(x)*5*x^2*e^(3*x)

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, 5*x^2 and e^(3*x) apply the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the parts of the product. Let u=5*x^2 and v=e^(3*x)

3. Differentiate each part individually. The derivative of u is (d(5)*x^2)/d(x)=10*x The derivative of v requires the chain rule: d(e^(3*x))/d(x)=e^(3*x)⋅(d(3)*x)/d(x)=3*e^(3*x)

4. Substitute these components into the product rule formula.

d(y)/d(x)=5*x^2⋅3*e^(3*x)+e^(3*x)⋅10*x

5. Simplify the expression by multiplying the terms.

d(y)/d(x)=15*x^2*e^(3*x)+10*x*e^(3*x)

6. Factor out the greatest common factor, which is 5*x*e^(3*x)

d(y)/d(x)=5*x*e^(3*x)*(3*x+2)

Final Answer

d()/d(x)*5*x^2*e^(3*x)=5*x*e^(3*x)*(3*x+2)

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