finding the second derivative?

Question

h(x) = SQRT(x^2 + 2)

h'(x) = x / SQRT(x^2 + 2) --> I think this is correct

In order to find the second derivative, I know I will need to use the Chain Rule and the Product Rule; but I am not sure how to implement this correctly. Thanks....

Answer 1

For this, it's easier to see it if you use fractional exponents. Rewrite h(x) like this:

h(x) = (x^2 + 2)^1/2
h'(x) = [(1/2)(x^2 + 2)^(-1/2)]2x
h'(x) = x(x^2 + 2)^(-1/2)

So your first derivative is right - nice job! Finding the second is going to be exactly what you said - product rule and chain rule. So it'd be first * derivative of second + second * derivative of first. The "derivative of second" part would require your chain rule, similar to what you did for the first derivative. So here we go:

h'(x) = x(x^2 + 2)^(-1/2)
h"(x) = x[(-1/2)(x^2 + 2)^(-3/2)](2x) + (x^2 + 2)^(-1/2)
h"(x) = (-x^2)(x^2 + 2)^(-3/2) + (x^2 + 2)^(-1/2)

You could try to factor and combine at this point if that was required, but this is also acceptable as an answer.