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If $ g $ is a twice differentiable function and $ f(x) = xg(x^2), $ find $ f" $ in terms of $ g, g', $ and $ g". $

$f^{\prime \prime}(x)=6 x g^{\prime}\left(x^{2}\right)+4 x^{3} g^{\prime \prime}\left(x^{2}\right)$

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in this problem, we're going to find F double prime of X. So of course we have to find f prime first. So the first derivative would be using the product rule. We have the first X times, the derivative of a second, and to find the derivative of G of X squared, we need to use the chain rule. So the derivative of the outside would be g prime of X squared times the derivative of the inside two x. So so far we have half the product rule done. We have the first times a derivative of the second. Now we have plus the second G of X squared times, the derivative of the first, which is just one the derivative of X is one. Okay, let's simplify that a little bit. So we have two x squared times g prime of X squared plus g of X squared, and we're going to find the derivative of this and that will be our second derivative. So f double prime notice. We have a product again. So we're going to use the product rule on that. So we have the 1st 2 x squared times, the derivative of the second. So the derivative of J Prime would be G double prime of X squared times, the derivative of the inside two x plus. The second g prime of X squared times a derivative of the first, The derivative of two X squared would be four x So all of this was the product rule on the first part. Now we still have plus the derivative of the second part, and the derivative of G would be g prime of X squared times, a derivative of the inside two x. Okay, let's see what we can do to simplify. So in the first term, we have two x squared times two x will multiply those and that's four x cubed with four x cubed times G double prime of X squared and then our second term is four x 10 g prime and our third term is two x times g prime. So we can add the forex and the two X and we get six x times g prime of X squared. Now the problem says we can give our answer in terms of G prime and G so we can leave it like that

Oregon State University