Problem

Use logarithmic differentiation to find the deriv…

02:44

Problem 46 Easy Difficulty

Use logarithmic differentiation to find the derivative of the function.

$ y = (\sqrt x)^x $

Answer

See explanation for answer

Watch More Solved Questions in Chapter 3

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

Problem 56

Video Transcript

in this problem. We are practicing, taking the derivative of a function using log arrhythmic differentiation, and that's going to make finding the derivative of the function were given much easier than a different way that we might use, like the limit definition or chain rule. So we're given the function. Why equals the square root of X rays to the X. Now we can simplify this by saying that this is the same thing is saying X rays to X over to. So the first thing that we're going to do is we'll take the natural log of each side and this is going to help us because we can move the exponents very easily using log properties. So we'll take the natural log of each side. We'll get the natural log of y equals the natural log of X rays to X over to. So now we can bring out that X over to term to get the natural log of y equals X over two times the natural log of X. Now, this is really nice, because we can use product rule now so well, find the derivative on each side will have won over why times by prime equals one half times the natural log of X plus X over two times one over X And again we found this side by using product rule. Now we could simplify that to say that one over. Why times y prime equals one half times and natural log of X plus one half. And now we need why prime by itself. So we would multiply by Why? To get white prime equals y over two times one plus the natural log of X. Now the only thing we have left to do is plug in. Why we know what why is that is the function that we started with. So why prime would be equal to the square root of X raised the X over two times one plus the natural log of X and that is our derivative of the function. I hope that this problem helped you understand a little bit more about lag, arrhythmic differentiation and why we use it and how that you could go through the process of using log properties and differentiation rules. In this case, the product rule

Madi S.

University of Denver

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 6

Derivatives of Logarithmic Functions

Problem