Find dy/dx y=x^3e^x

Problem

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

Solution

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

2. Assign the functions to the variables u and v Let u=x^3 and v=e^x

3. Differentiate each part individually. The derivative of u is d(x^3)/d(x)=3*x^2 and the derivative of v is d(e^x)/d(x)=e^x

4. Apply the product rule formula by substituting the functions and their derivatives.

d(y)/d(x)=x^3d(e^x)/d(x)+e^xd(x^3)/d(x)

5. Substitute the calculated derivatives into the expression.

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

6. Simplify the expression by factoring out the common terms x^2 and e^x

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

Final Answer

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

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