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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.$$y=\ln \left(x^{3} \sqrt{3 x+1}\right)$$

$$\frac{3(7 x+2)}{2 x(3 x+1)}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

Missouri State University

Campbell University

Idaho State University

Lectures

01:46

Use the properties of loga…

02:07

03:11

02:15

01:22

02:22

03:09

Use logarithmic differenti…

01:18

02:09

01:11

eso This problem is set up really nicely with the fact that you're already doing the natural log three x plus one. So I mean, try Thio. I mean, I can split this up. Eso splitting it up makes the problem a lot easier. So we have natural you execute because it's multiplication changes to plus natural log. Now this is my second step is well that most students air pretty familiar that the square root is the same thing as to the one half power. Now I think students could also, um, do the power rule at the same time which is moving that exponents in front. So we're looking at why equals three natural log of X plus one half natural log of three X plus one. Now if you can't, there's a step by step process there toe understanding. So now we're ready to take the derivative because when you have a constant times, some function that constant is just going along for the ride. The derivative of natural log of X is one of her ex same thing. One half is a constant, so just going along for the ride the derivative of natural log is one over that piece. But don't forget the chain rule. You have to also take the derivative of the inside. The derivative of three X plus one is three. Now, if I were cleaning this up, I would just leave it as three over X plus one times one times three is three over. I would leave as two times a quantity of three X plus one. Now, if you wanted to simplify this, you could. But I don't see any reason to doing that. Um, yeah, This this answer should be fine. It's just good enough, okay?

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