49–55. Derivatives of tower functions (or g^h) Find the deriva...

Question

49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.

g (x) = x^ In x; a = e

Accepted Answer

Determine the derivative of H X equals 2 X, raise the two natural log of X, and evaluate H E. Now, to solve this, we'll actually make use of the log derivative. So Well, first rewrite our function. In Y equals 2 X C 2 natural lax. Now, to do the log derivative, we'll need to take natural log of both sides. Giving natural luck Y equals natural luck. 2 X raise to the 2, natural log X. And because we have a product inside of our log function, we can rewrite it as natural log of 2, plus natural log X race the 2, natural log X. Now, let's apply one of our log properties. This property states that if we have log. Of some raised to an exponent, we can rewrite it as the exponent multiplying the log of The base In other words, natural log X to the A is equal to a natural log of X. We can use this in our function. Or this natural log X raced the two L and X. We get 2 natural log X, multiplied by natural log of X. Giving us natural log of 2 + 2 natural log of X squared. Now we can take our derivatives. So, for our derivative, we'll take the derivative of the left side to be 1 divided by Y. Why It equals 2. 0 plus. 2, multiplied by 2 natural log X. Multiplied by 1 divided by X. So, now, a breakdown of this. Our left side here, natural log of Y, the derivative of natural log is 1 divided by Y. So we get one divided by Y. Now this is an implicit derivative on the left, which means you multiply this by a Y. On the right side, the derivative of the natural log of 2 is 0, because that's a constant. We then have the derivative of natural log of X2. Which is a chain rule. So, natural log of X2 derivative is to natural log of X. And the inside function, natural log of X's derivative is one divided by X. Now we can simplify. 21 divided by Y Y is equals to 4, natural log of X divided by X. Now, if we want Y by itself, we notice that we have 1 divided by Y. We can multiply Y on both sides. To get Y by itself. We get Y is equals to 4 natural log of X divided by X, multiplied by Y. Now, if we look back at our starting function, we claimed that Y was equals to 2 X raises to the two natural log of X. We can make a substitution right here for why. We get Y prime. Equals for natural log X divided by X, multiplied by 2 X raises to the 2 natural log of X. We can then simplify To get 8, natural lag of X. Multiplied by X raises to the two natural log X, all divided by X. And finally We can combine The X and the denominator. With the X and the numerator by an exponent rule. Where you have X, raises some power divided by X to another power is equivalent to X. Of the two powers the difference. In other words, X to the A divided by X B equals X raises to the A minus B. We get Y is equals to 8 X2, natural lax. -1 Multiplied by natural log of X. Now, we can plug in our value of H. Which is E in our case. So I will go over here and write. H prime of E And plug in E for all of our X values. 8 E raise the two. Natural lag E minus 1. Multiplied by natural lag of E. Now, natural log of E cancels itself out to get 1. So we get 8 E rated to the 2 multiplied by 1 minus 1, all multiplied by 1. Which gives us just 8 e to the first power. Making our final answer 8 E. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.
g (x) = x^ In x; a = e

Key Concepts

Derivatives

Exponential and Logarithmic Functions

Chain Rule

Watch next