Find d/dx(ln(x/x²+1)) without using the Quotient Rule.

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Welcome back, everyone. In this problem, we want to determine the derivative of Y equals LN 3X divided by 2X2 + 4 with respect to X without using the quotient rule. A says it's X2 + 2 divided by X2 minus 2, B 2 minus X2 divided by X2 + 2. C 2 minus X2 divided by X multiplied by X2 + 2 and D X + 2 divided by X multiplied by X2 minus 2. Now if we're going to determine the derivative of Y without using the quotient rule, where obviously here we have a caution in our expression, then we'll need to use some property of log to rewrite our natural log in a form that's not a caution. So how can we do that? Well, first, let's ask ourselves what do we know about quotient in logs. Recall, OK. That The natural log LN of a quotient A divided by B is the same thing as LNA. Minus LNB. In this way, it's not written as a quotient. So if we apply this property of logs to our problem, that means we could rewrite Y then as LN 3X minus LN2X2 + 4. And know that it's not a ca we can determine the derivative with respect to x. Now what do we know about the derivatives of locks of natural locks? Well, recall, OK, that the derivative, OK, of the natural log of FF X, some function with respect to X, is going to be equal to the reciprocal of FF X, OK, multiplied by the derivative of FF X. In other words, we differentiate the function inside the log, inside the natural log, and then we divide it by that original function. So in this case then, This would imply that the derivative. Of the natural log of 3 X, OK. Is going to be equal to 1 divided by 3 X, multiplied by the derivative of 3 X which is 3. And when we simplify that, it leaves us with 1 divided by X. Likewise here, OK, when we differentiate. Our second term, so that's the natural log of 2 X2 + 4, OK. Then it's gonna be 1 divided by 2 X2 + 4 multiplied by the derivative of 2 X2 + 4, which is 4 X. Now, if we go ahead and simplify there, we can factor or 2 from our denominator. OK. And when we factor 2 from the numerator and denominator, we're left with 2 X divided by X2 + 2. Know that we have the derivatives of both our terms, then we can find the derivative of Y because no, that means the derivative of Y is going to be 1 divided by X minus 2 x divided by X2 + 2. If we combine these fractions, OK, then it's gonna be X2 + 2 minus 2 X2, OK? All divided by X multiplied by X2 + 2. And no, take it a step further, X2 minus 2x2 equals negative X2. So the derivative of Y then will be 2 minus X2 divided by X multiplied by X2. Plus 2 So that means then the derivative of Y is X2 + 2 sorry. 2 minus X2 divided by X multiplied by X2 + 2. And when we look back on our answer choices, that tells us C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Find d/dx(ln(x/x²+1)) without using the Quotient Rule.

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