In Exercises 1–24, find the derivative of y with respect to th...

Question

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

13. y = (x+2)^(x+2)

Accepted Answer

Welcome back, everyone. Use logarithmic differentiation to find D Y divided by D X or Y equals X + 3 to the power of X + 3. For this problem, let's begin by taking the natural log of both sides because it says that we want to use logarithmic differentiation. So Llan of the left hand side, a line of Y, is going to be equal to Llan of the right hand side, or basically a line of X + 3 to the power of X + 3. This is particularly useful because we can now apply the power rule of logarithms. The left hand side remains unchanged, and on the left hand side, we, I'm sorry, on the right hand side, we get X + 3, we can bring down the exponent multiplied by LN of X + 3. And from here we can differentiate both sides. On the left hand side, we are going to get the derivative of LN of Y. We have to apply the chain rule because Y is a function of X, so we're starting with 1 divided by Y and we are multiplying by Y, which is our derivative. So, here we have applied to the chain rule, right? On the right hand side, we have a product of two functions. Let's remember how we differentiate a product. We take the derivative of the first function. Multiplied by the 2nd function. And we add the first function. Which is multiplied by the derivative of the 2nd 1. So what we want to do now is identify the remaining derivatives and solve for Y. For the remaining derivatives, well, we can show that the derivative of X + 3 is equal to 1, the derivative of X is 1, the derivative of 3 is 0, and for the derivative of natural log of X + 3, we want to apply the chain rule. So we are starting with 1 divided by X + 3 when we differentiate LN with respect to X + 3. And according to the chain rule, we're multiplying by the derivative of X + 3. We already know that the derivative of X + 3 is 1, so we can basically cross it out and show that it is equal to 1. So now what do we get? Well, on the left hand side, that's Y divided by Y. On the right A line of x + 3. Plus X + 3 multiplied by 1 divided by X + 3. So, X + 3 divided by X + 3 can be simplified. That's basically 1. And now we are ready to solve for Y prime, which is our derivative. Let's multiply both sides by Y, so we get Y equals Y multiplied by LN of X + 3 + 1. And we also know that Y is equal to X + 3. To the power of X + 3. So we can now replace Y with that expression and multiply by the factor in parent, which is alan of X + 3 + 1. Well then, this is how we applied logarithmic differentiation to get the derivative of Y. Thank you for watching.

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.13. y = (x+2)^(x+2)

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In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
13. y = (x+2)^(x+2)