Find the Derivative - d/dx y=x^3e^(4x)

Problem

d()/d(x)*x3*e(4*x)

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x3 and e(4*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=x3 and v=e(4*x)

3. Differentiate each part separately. The derivative of u is d(x3)/d(x)=3*x2 The derivative of v requires the chain rule: d(e(4*x))/d(x)=e(4*x)⋅(d(4)*x)/d(x)=4*e(4*x)

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

d(y)/d(x)=x3*(4*e(4*x))+e(4*x)*(3*x2)

5. Simplify the expression by factoring out the greatest common factor, which is x2*e(4*x)

d(y)/d(x)=x2*e(4*x)*(4*x+3)

Final Answer

(d(x3)*e(4*x))/d(x)=x2*e(4*x)*(4*x+3)

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