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

Problem

d2()/(d(x)2)*x*(x−4)3

Solution

1. Identify the function as ƒ(x)=x*(x−4)3 and note that finding the second derivative requires applying the product rule twice or expanding the expression.

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=(x−4)3

3. Differentiate the components: d(x)/d(x)=1 and d(x−4)/d(x)=3*(x−4)2 using the chain rule.

4. Calculate the first derivative:

(d(x)*(x−4)3)/d(x)=x⋅3*(x−4)2+(x−4)3⋅1

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

(d(x)*(x−4)3)/d(x)=(x−4)2*(3*x+x−4)

(d(x)*(x−4)3)/d(x)=(x−4)2*(4*x−4)

(d(x)*(x−4)3)/d(x)=4*(x−1)*(x−4)2

6. Apply the product rule again to find the second derivative of 4*(x−1)*(x−4)2 Let u=4*(x−1) and v=(x−4)2

7. Differentiate the new components: (d(4)*(x−1))/d(x)=4 and d(x−4)/d(x)=2*(x−4)

8. Calculate the second derivative:

(d2(x)*(x−4)3)/(d(x)2)=4*(x−1)⋅2*(x−4)+(x−4)2⋅4

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

(d2(x)*(x−4)3)/(d(x)2)=4*(x−4)*(2*(x−1)+(x−4))

(d2(x)*(x−4)3)/(d(x)2)=4*(x−4)*(2*x−2+x−4)

(d2(x)*(x−4)3)/(d(x)2)=4*(x−4)*(3*x−6)

10. Factor out a 3 from the second binomial:

(d2(x)*(x−4)3)/(d(x)2)=4*(x−4)⋅3*(x−2)

(d2(x)*(x−4)3)/(d(x)2)=12*(x−4)*(x−2)

Final Answer

(d2(x)*(x−4)3)/(d(x)2)=12*(x−4)*(x−2)

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