Find the 2nd Derivative y=xe^x

Problem

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

Solution

1. Identify the function y=x*e^x and note that finding the second derivative requires applying the product rule twice.

2. Apply the product rule for the first derivative, where (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) Let u=x and v=e^x

3. Calculate the first derivative d(y)/d(x)

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

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

4. Differentiate again to find the second derivative d^2(y)/(d(x)^2) by applying the sum rule and the product rule to the result of the first derivative.

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

5. Substitute the derivative of x*e^x (which was already found in step 3) and the derivative of e^x

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

6. Simplify the expression by combining like terms.

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

7. Factor out the common term e^x to reach the final form.

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

Final Answer

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

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