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

Problem

d()/d(x)*5*x2*e(3*x)

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, 5*x2 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*x2 and v=e(3*x)

3. Differentiate each part individually. The derivative of u is (d(5)*x2)/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*x2⋅3*e(3*x)+e(3*x)⋅10*x

5. Simplify the expression by multiplying the terms.

d(y)/d(x)=15*x2*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*x2*e(3*x)=5*x*e(3*x)*(3*x+2)

Want more problems? Check here!