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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.$$y=x^{3} 4^{2 x}$$

$$x^{2} 4^{2 x}(2 x \ln 4+3)$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

Missouri State University

Baylor University

Idaho State University

Lectures

02:15

Use the properties of loga…

03:11

02:55

02:07

01:46

02:22

01:11

01:59

02:09

05:15

Use logarithmic differenti…

01:22

01:18

01:50

uh huh. To do this problem, we're gonna just take the natural aga both sides x cubed forward to the two X But I just want to rewrite that problem because that's gonna be important later. So as we take the natural aga both sides, what we can do is we can split up the right side. Okay, so we have a multiplication in here so we can rewrite it as natural log of X cubed plus natural log of four to the two x on. We do that because we can move these powers in front. Um, so we're looking at three natural log of X. Now I'm going to rewrite It is two x natural law before, but I just like to see a constant Times X, uh, in that problem now, we still have natural log of why on the left side. So as I take the derivative, we have one over why? This is implicit differentiation, by the way. And right here we have three times because three is a constant. Yeah, the derivative of natural log of X is one over X. And then the derivative of this piece is a constant Times X. Well, a constant times X derivative is the constant. So it's just that to natural law before. But we need to solve for d y d x on we solve for d y d x By multiplying this y over now, we started without any excess in the problem So we wanna end without any excess in the problem. We're sorry with Onley X is we don't want this. Why over here is my point. Eso What's why equal to at the beginning we established that why was equal to this X cubed forward to the two X so I can replace this Why with X cubed four to the two X Now if you want to, you could distribute this in, you know, to both terms but you're just wasting your time. This answer is good enough and we can stop right there. This is good

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