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Find the derivative of the function.

$ y = x^2 e^{-3x} $

$y^{\prime}=x e^{-3 x}[2-3 x]$

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here's our function and we're going to find its derivative and this function is actually a product. So the first factor is X squared, and the second factor is e to the negative three x. So we're going to need to use the product rule to differentiate. But we're going to find that in the midst of doing the product rule will stop and do the chain rule. So let's find our derivative. Why prime? So according to the product rule, we have the first function X squared times, the derivative of the second. Now here's where we're going to need to use the chain rule because each of the negative three x is a composite, It's outside function is the e to the X function, and it's inside function is negative. Three x. So the derivative of the outside would give us e to the negative three x times The derivative of the inside the derivative of negative three X would be negative. Three. So what we have so far is the first times the derivative of the second, continuing with the product rule now plus the second e to the negative three x times the derivative of the first. So times the derivative of X squared and that would be two x. So we had the first times driven about second plus, the second time's a derivative of the first. Okay, so we have our derivative and now we're going to simplify. So we have basically two terms. We have this first term, a product of several things, and we have this last term, a product of several things. Do you notice that there are some common factors. Both of these terms have an X that we could factor out. And both of these terms have an E to the negative three X that we could factor out. So if we do that, we have x times e to the negative three x times. What do we have left in this first term? We have the negative three and we have one of the exes. So we have negative three X And what do we have left in the last term? We have the two, so we have plus two. Okay, now the only other thing we might choose to do with this is just to rewrite negative three x plus two as two minus three x because that looks a little bit nicer. So are derivative is x Times E to the negative. Three X Times two minus three x

Oregon State University