Find the 2nd Derivative y=x(x-4)^3

Problem

d2()/(d(x)2)*x*(x−4)3

Solution

1. Identify the function as y=x*(x−4)3 and recognize that finding the second derivative requires applying the product rule and the chain rule twice.

2. Apply the product rule for the first derivative, where d()/d(x)*[u*v]=ud(v)/d(x)+vd(u)/d(x) Let u=x and v=(x−4)3

d(y)/d(x)=x⋅d(x−4)/d(x)+(x−4)3⋅d(x)/d(x)

3. Differentiate the components using the chain rule for (x−4)3

d(y)/d(x)=x⋅3*(x−4)2*(1)+(x−4)3*(1)

4. Simplify the first derivative by factoring out the common term (x−4)2

d(y)/d(x)=(x−4)2*[3*x+(x−4)]

d(y)/d(x)=(x−4)2*(4*x−4)

d(y)/d(x)=4*(x−1)*(x−4)2

5. Apply the product rule again to find the second derivative d2(y)/(d(x)2) Let u=4*(x−1) and v=(x−4)2

d2(y)/(d(x)2)=4*(x−1)⋅d(x−4)/d(x)+(x−4)2⋅(d(4)*(x−1))/d(x)

6. Differentiate the components.

d2(y)/(d(x)2)=4*(x−1)⋅2*(x−4)*(1)+(x−4)2⋅4

7. Simplify the expression by factoring out 4*(x−4)

d2(y)/(d(x)2)=4*(x−4)*[2*(x−1)+(x−4)]

d2(y)/(d(x)2)=4*(x−4)*[2*x−2+x−4]

d2(y)/(d(x)2)=4*(x−4)*(3*x−6)

8. Factor out a 3 from the second binomial to reach the final form.

d2(y)/(d(x)2)=12*(x−4)*(x−2)

Final Answer

d2()/(d(x)2)*x*(x−4)3=12*(x−4)*(x−2)

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