Find the 2nd Derivative y=xe^x

Problem

d2()/(d(x)2)*x*ex

Solution

1. Identify the function y=x*ex 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=ex

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

d(y)/d(x)=xd(ex)/d(x)+exd(x)/d(x)

d(y)/d(x)=x*ex+ex

4. Differentiate again to find the second derivative d2(y)/(d(x)2) by applying the sum rule and the product rule to the result of the first derivative.

d2(y)/(d(x)2)=(d(x)*ex)/d(x)+d(ex)/d(x)

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

d2(y)/(d(x)2)=(x*ex+ex)+ex

6. Simplify the expression by combining like terms.

d2(y)/(d(x)2)=x*ex+2*ex

7. Factor out the common term ex to reach the final form.

d2(y)/(d(x)2)=ex*(x+2)

Final Answer

d2()/(d(x)2)*x*ex=ex*(x+2)

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