Find the 2nd Derivative 12x(x^2+10)^5

Problem

d2()/(d(x)2)*12*x*(x2+10)5

Solution

1. Identify the function as ƒ(x)=12*x*(x2+10)5 and prepare to find the first derivative using the product rule and chain rule.

2. Apply the product rule for the first derivative, where u=12*x and v=(x2+10)5

d(ƒ(x))/d(x)=12*(x2+10)5+12*x⋅5*(x2+10)4*(2*x)

3. Simplify the first derivative expression by combining terms.

d(ƒ(x))/d(x)=12*(x2+10)5+120*x2*(x2+10)4

4. Factor out the common term 12*(x2+10)4 to make the second derivative easier to calculate.

d(ƒ(x))/d(x)=12*(x2+10)4*(x2+10+10*x2)

d(ƒ(x))/d(x)=12*(x2+10)4*(11*x2+10)

5. Apply the product rule again to find the second derivative, using u=12*(x2+10)4 and v=11*x2+10

d2(ƒ(x))/(d(x)2)=48*(x2+10)3*(2*x)*(11*x2+10)+12*(x2+10)4*(22*x)

6. Simplify the terms in the second derivative.

d2(ƒ(x))/(d(x)2)=96*x*(x2+10)3*(11*x2+10)+264*x*(x2+10)4

7. Factor out the common term 24*x*(x2+10)3 to reach the final simplified form.

d2(ƒ(x))/(d(x)2)=24*x*(x2+10)3*[4*(11*x2+10)+11*(x2+10)]

d2(ƒ(x))/(d(x)2)=24*x*(x2+10)3*(44*x2+40+11*x2+110)

d2(ƒ(x))/(d(x)2)=24*x*(x2+10)3*(55*x2+150)

8. Factor out a 5 from the last binomial to finish simplification.

d2(ƒ(x))/(d(x)2)=120*x*(x2+10)3*(11*x2+30)

Final Answer

d2()/(d(x)2)*12*x*(x2+10)5=120*x*(x2+10)3*(11*x2+30)

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