Find the 2nd Derivative f(x)=(x^2+8)^9

Problem

d2()/(d(x)2)*(x2+8)9

Solution

1. Identify the function ƒ(x)=(x2+8)9 and recognize that finding the second derivative requires applying the chain rule twice.

2. Apply the chain rule to find the first derivative ƒ(x)′ by differentiating the outer power and multiplying by the derivative of the inner function x2+8

ƒ(x)′=9*(x2+8)8⋅d(x2+8)/d(x)

ƒ(x)′=9*(x2+8)8⋅(2*x)

ƒ(x)′=18*x*(x2+8)8

3. Apply the product rule to ƒ(x)′=18*x*(x2+8)8 to find the second derivative ƒ(x)″ where the two functions are 18*x and (x2+8)8

ƒ(x)″=(d(18)*x)/d(x)⋅(x2+8)8+18*x⋅d(x2+8)/d(x)

4. Differentiate each part, using the chain rule again for the second term.

ƒ(x)″=18*(x2+8)8+18*x⋅(8*(x2+8)7⋅2*x)

ƒ(x)″=18*(x2+8)8+18*x⋅(16*x*(x2+8)7)

ƒ(x)″=18*(x2+8)8+288*x2*(x2+8)7

5. Factor out the greatest common factor, which is 18*(x2+8)7

ƒ(x)″=18*(x2+8)7⋅((x2+8)+16*x2)

6. Simplify the expression inside the parentheses.

ƒ(x)″=18*(x2+8)7*(17*x2+8)

Final Answer

d2()/(d(x)2)*(x2+8)9=18*(x2+8)7*(17*x2+8)

Want more problems? Check here!