In Exercises 115–126, use logarithmic differentiation or the m...

Question

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.

126. eʸ = y^(ln x)

Accepted Answer

Welcome back everyone. Use logarithmic differentiation to find the derivative of Y with respect to X or E to the power of 3 Y equals Y to the power of LN X. For this problem, let's remember that logarithmic differentiation begins by taking the natural log of both sides of our equation. So we begin with LN of e to the power of 3 Y and set it equal to Llan of the right hand side, or LN of Y to the power of LN of X. You know, this is particularly useful because we can apply the power rule of logarithms. On the left-hand side, we can bring down the exponents. We get 3 Y multiplied by LN of E. On the right hand side, we can bring down Ean of X. Which is now multiplied by LN of Y. Let's also remember that Ean of E is equal to 1. So we can simplify this expression and show that V Y is equal to L N X multiplied by LN of Y. And we are ready to differentiate both sides of this equation. On the left we can factor out 3. And the derivative of Y can be written as Y using a shorthand notation. On the right hand side we have a product of two functions, so we want to apply the product rule. We take the derivative of the first function and multiply by the 2nd 1, and we add the first function multiplied by the derivative of the 2nd 1. So now our goal is to identify the remaining derivatives. We get 3 Y equals. The derivative of LN X is a 1 divided by X, so we can just write LN of Y divided by X. Plus LN of X is multiplied by the derivative of LN of Y, so this one requires chain rule because Y is a function of X. We first differentiate LN of Y with respect to Y, which gives us 1 divided by Y, and then we multiply by Y. That's because Y is a function of X. We want to multiply by Y based on the chain rule. So now, let's Write the results on form. We get 3 Y on the left. On the right, LN of Y is divided by X, and we are adding. Y multiplied by LNX divided by Y. Remember that our goal is to solve for a Y, so we can bring the Y terms on the left hand side. We're going to get 3 Y minus Y. LN of X divided by Y. Equals the remaining term on the right is Llan of Y divided by X. From here we can factor out Y. In priss we're going to get 3. Minus LN of X divided by Y. Equals lm of Y divided by X. So now solving for Y, we're going to take our right hand side. Which is LN of Y divided by X and divided by the factor on the left hand side. That's 3 minus. LN of X. Divided by Why, right? And from here, we basically want to simplify because we have two denominators, X and Y. Let's just multiply both sides by XY. So, Y is going to be equal to XY divided by X gives us Y in the numerator, so we're going to get Y LN of Y. In the denominator, 3 multiplied by X Y is 3 XY. And we are subtracting L of X divided by Y. Multiplied by X Y. So, we basically got XY divided by Y, which is X, and this means that we're subtracting XLM X. We can also factor out and show that Y is equal to Y LN of Y. Divided by Factoring out X, we get X and C 3 Y minus. Alan of X, which is our final answer for this problem. Thank you for watching.

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.126. eʸ = y^(ln x)

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In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
126. eʸ = y^(ln x)