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

Problem

d^2()/(d(x)^2)*12*x*(x^2+10)^5

Solution

1. Identify the function as ƒ(x)=12*x*(x^2+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=(x^2+10)^5

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

3. Simplify the first derivative expression by combining terms.

d(ƒ(x))/d(x)=12*(x^2+10)^5+120*x^2*(x^2+10)^4

4. Factor out the common term 12*(x^2+10)^4 to make the second derivative easier to calculate.

d(ƒ(x))/d(x)=12*(x^2+10)^4*(x^2+10+10*x^2)

d(ƒ(x))/d(x)=12*(x^2+10)^4*(11*x^2+10)

5. Apply the product rule again to find the second derivative, using u=12*(x^2+10)^4 and v=11*x^2+10

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

6. Simplify the terms in the second derivative.

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

7. Factor out the common term 24*x*(x^2+10)^3 to reach the final simplified form.

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

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

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

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

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

Final Answer

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

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