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Numerade Educator

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Problem 36 Hard Difficulty

Find the derivative of the function.
$ y = x^2 e^{-1/x} $

Answer

$\frac{d y}{d x}=e^{-1 / x}(2 x+1)$

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Video Transcript

here we have a function that's a product. So we're going to end up using the product rule to differentiate it. And before I start that, I'm going to rewrite the negative one over X power. So we're gonna write this as X squared times e raised to the negative X to the negative one. That'll make it easier to differentiate. Okay, so for the derivative, we're going to use the product rule. So we start with the first X squared times, the derivative of the second. So now it's time to take the derivative of E to the negative x the negative one power. So we would start with E to the negative X to the negative one power and then multiply by that power. And so the derivative of negative X to the negative one would be positive one times X to the negative, too. Because you bring down the negative one, multiply it by the negative that's already there. And then you subtract one to get the new power. Okay, so so far, we have the first times the derivative of the second. Now it's going to be plus the second e to the negative X to the negative one times the derivative of the first and the derivative of the first would be X squared two X derivative of X Squared is two X. Now let's see what we can do with this. So notice that we have X squared times X to the negative. Second, if you add the powers on X there you get X to the zero and X to the zero is one. So that term is actually just e to the negative X to the negative one. And then the other term is two x times e to the negative x the negative one So we can factor e to the negative X to the native one out of both of those terms, and it's time to rewrite it back in its original form E to the negative one over X that's being factored out. So what we have left in the first term is just one on what we have in the other term is two x So we one plus two x times e to the negative one over x