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Use the properties of logarithms to find the derivative. Hint: It might be easier if you use the properties of logarithms first.$$y=\frac{4^{-2 x^{2}}}{x^{2}+1}$$

$$\frac{-2 x 4^{-2 x^{2}}\left(2\left(x^{2}+1\right) \ln 4+1\right)}{\left(x^{2}+1\right)^{2}}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

Campbell University

Baylor University

University of Michigan - Ann Arbor

Lectures

02:22

Use the properties of loga…

03:11

02:15

01:11

01:50

Use logarithmic differenti…

01:22

02:07

01:46

03:01

01:14

right. Alright. This problem is gonna take me a while, so I'm just gonna jump into the set up. Yeah, Forwards. Negative two x squared. Yeah, over X squared plus ones Eso The way to do this problem is the natural log both sides. The reason why you want a natural log both sides is because with a for two the name two x square over X squared plus one s So then you can split up the log because of the division in here with subtraction. Now I'm also going to do another step is well because instead of writing, um, each one individually and something else What I also want to do is acknowledge that that exponents, once you have into a log, can be moved in front. Eso We're looking at negative two x squared eso. The important things to remember is that natural log A four is just a constant. That number is always the same. So when I take the derivative of this piece, it's gonna be negative. Four x in that natural log of four is still that constant. Um, I like to write it this way to help soon, recognize? But I do realize that a lot of people, right? The X right here. So there's less confusion, but that's a personal preference. Um, whereas a derivative on the right side, you have won over you leave the function alone. But then you have to multiply by the derivative of the inside, which is two X And then don't forget the implicit differentiation on this side is one over Y D Y d x And you also heard me say is gonna take a while because now, once you simplify that you have to solve a d y the x by multiplying the why over eso Our answer for D y d x is Ah yeah, Let's go ahead and leave this. You know the way A lot of people, right? Nega Forex National Aga four minus two X over X squared plus one. But then times by why and and we define why, in the beginning of the problem is forward to the native two x squared, Yeah, over X squared plus one. Now, if you wanted to, you could simplify this. You know, you could distribute this piece into both terms, but I don't see much benefit of doing that. Um, the answer key probably does, but this is good enough. What? What I have circled is good enough

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