In Exercises 7–38, find the derivative of y with respect to x,...

Question

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.

23. y = ln(x)/(1+ln(x))

Accepted Answer

Welcome back, everyone. Find the derivative of Y with respect to X or Y equals LNX divided by 2 + LN X. For this problem, let's recognize that we have a function which is a rational function, right? We have a function of X in the numerator and a function of X in the denominator. So to differentiate it, we have to apply the quotient rule. Let's remember that first we take the derivative of the numerator and we multiply by the function in the denominator. We're subtracting the function in the numerator multiplied by the derivative of the function in the denominator. And then we are dividing by the square. Of the denominator, so that'll be 2 plus LN of X squared. Now, let's recognize that we want to identify two derivatives. One of them is LN of X. It is equal to 1 divided by X. And then we have 2 plus LNX derivative, the derivative of 20. So once again, we want to find the derivative of LN of X, which is 1 divided by X. And then we can substitute into the original quotient rule. We're going to get 1 divided by X multiplied by 2 plus LN of X. Minus LN of X multiplied by 1 divided by X. And this is divided by 2 plus LN of X squared. Now, what we can do is simply multiply both sides by X, right? Both our numerator and our denominator. So in the denominator, we will have X multiplied by 2 plus LNX squad, and then considering the remaining terms in the numerator, we're going to have 2 + LNX minus LN X. So that's simply 2, meaning the final derivative is. 2 divided by X, which is multiplied by 2 plus LN of X squared. Thank you for watching.

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.23. y = ln(x)/(1+ln(x))

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In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
23. y = ln(x)/(1+ln(x))