Find d/dx (In(xe^x)) without using the Chain Rule and the...

Question

Find d/dx (In(xe^x)) without using the Chain Rule and the Product Rule.

Accepted Answer

Welcome back, everyone. In this problem, we want to find the derivative of the function F X E equals LN 2 X E 2 X. A note is that we're not supposed to apply the chain or product rule of differentiation. A says the derivative is 2 X + 1 divided by X, B, 2 X + 1 divided by X squared, C 2 X divided by X + 1, and D X + 2 divided by X. Now, if we're not going to apply the chain or product rules, that means we'll have to apply the properties of logarithms instead to rewrite FFX in a convenient way that we can differentiate without them. So how can we do that? What do we know here? Well, notice that here we're finding the natural log of a product. And recall, OK, that if we're finding the log, OK, of something like that where it says FGH of X. We could rewrite it as the natural log or or the log rather of FF X plus the log of G of X. OK, let me write that properly here. Plus the log of G of X, in this case our log is the natural log LN plus the natural log here of H of X, OK. Now in this case, what do we notice for our function? Well, from the natural log of 2 XE to the 2 X here. Notice that We can tell we have one term here, F of X to be 2, another G of X to be X, and another H of X to be E 2 X. OK? Then that means applying that idea using that property of logs. Let me just clean this up a bit here. Using that property, then we could rewrite the natural log of 2 X E to the 2 X. As the natural lager to. Plus the natural log of X, plus the natural log of E to the 2 X. Now that we're able to write it this way, then we don't need to use the chain or product to, OK? Because we can differentiate each of these individually, OK? So, how are we going to do that? Well, notice here that first we're finding the natural log of 2, and this is a constant, so we know that will be 0. Next, the, the, well, let me write that properly here. We're finding the derivatives, sorry, not the actual logs. So I'll go 2 XE to the 2 X. OK. Now the derivative of the natural log of 2 is 0. The derivative of the natural log of X is going to be 1 divided by X, OK. And here, notice that if we're finding natural the derivative of LNE to the 2 X, we could basically rewrite this as the derivative of 2 X because we could bring down the power, OK? In other words, we could rewrite this as 2 X multiplied by the natural log of E which is gonna be 2 X multiplied by 1, which is 2 X. OK? So we could take all of this here and just write this as the derivative of 2 X. And then when we simplify that, this is gonna be 1 divided by X plus 2 or 2 + 1 divided by X. Now, to make it simpler, we could write it under a single fraction, and that's gonna be 2 X + 1 divided by X. Thus, this is gonna be the derivative of. Or function, OK? The derivative of LN2XE to the 2 X. The derivative of F of X. If we look back at our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Find d/dx (In(xe^x)) without using the Chain Rule and the Product Rule.

Key Concepts

Natural Logarithm Properties

Exponential Functions

Basic Differentiation Rules

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