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

Problem

d^2()/(d(x)^2)*(x^2+8)^9

Solution

1. Identify the function ƒ(x)=(x^2+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 x^2+8

ƒ(x)^′=9*(x^2+8)^8⋅d(x^2+8)/d(x)

ƒ(x)^′=9*(x^2+8)^8⋅(2*x)

ƒ(x)^′=18*x*(x^2+8)^8

3. Apply the product rule to ƒ(x)^′=18*x*(x^2+8)^8 to find the second derivative ƒ(x)^″ where the two functions are 18*x and (x^2+8)^8

ƒ(x)^″=(d(18)*x)/d(x)⋅(x^2+8)^8+18*x⋅d(x^2+8)/d(x)

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

ƒ(x)^″=18*(x^2+8)^8+18*x⋅(8*(x^2+8)^7⋅2*x)

ƒ(x)^″=18*(x^2+8)^8+18*x⋅(16*x*(x^2+8)^7)

ƒ(x)^″=18*(x^2+8)^8+288*x^2*(x^2+8)^7

5. Factor out the greatest common factor, which is 18*(x^2+8)^7

ƒ(x)^″=18*(x^2+8)^7⋅((x^2+8)+16*x^2)

6. Simplify the expression inside the parentheses.

ƒ(x)^″=18*(x^2+8)^7*(17*x^2+8)

Final Answer

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

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