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Solve the given problems by finding the appropriate derivatives.If $f(x)$ is a differentiable function, find an expression for the derivative of $y=x^{2} f(x)$.

$\frac{d y}{d x}=x^{2} f^{\prime}(x)+2 x f(x)$

Calculus 1 / AB

Chapter 23

The Derivative

Section 6

Derivatives of Products and Quotients of Functions

Derivatives

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In mathematics, a function…

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So these are some classic calculus problems that actually get you to make sure that you know the product rule, because the last students have figured out how to fake the product rule and things like that. Um So to correctly do this problem again, identify where your product is so that you do the product rule correctly. And a lot of times I'll even put in four blanks with a plus in here to help students, especially the first time they're learning it. So when you're tasked with finding the derivative dy dx, you have to take the derivative of one side. So I like to do the left side first times you leave the right side alone and then plus now you leave the side, you took the derivative of the X squared alone and then times the derivative of the right side or the other side. Um and we don't know anything about f just that it's differentiable. So we just write down the notation so this is a perfect answer. Remember additions communities. So you can, you can switch the order around if you really want to. I'm not sure why anybody would do this, but you could also, if you want to factor out an X and the problem. So you'd have two times ffx plus X times f prime of X. There's a lot of different ways that you can write the correct answer and it's really up to you what makes most sense to you or your teacher. So these are both correct and it utilizes the product rule.

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