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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.$$y=\ln \left(3 x^{4} \sqrt[5]{2 x^{2}+9}\right)$$

$$\frac{4\left(11 x^{2}+45\right)}{5 x(2 x+9)}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

McMaster University

Harvey Mudd College

Baylor University

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Use the properties of loga…

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Find the derivative of the…

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All right. So our task is to find the derivative of this function, which really does just look scary than what it really is. Three x to the fourth. And then we have the square root of I'm sorry. The fifth root of two x squared. Mhm plus sign. All right, so the whole premise of this is that you can rewrite this as I'm also gonna put a multiplication there as well is a natural log of three plus, because when you're multiplying inside of a log and split it up into, uh, addition of the fifth root Well, how about this? Instead of writing the fifth root calculus, students should know that the fifth root is the same thing as to the 1/5 power. So you have this two x squared plus nine. So now what I can do is I can just move these powers in front, so I'm kind of doing two steps at once, but I'm trying to show my work along the way. Um, now, another important thing to recognize. So this is a law of logs that I'm doing Is that natural log of three is always a constant, uh, and the drift of of a constant is always zero. So when I take the derivative of this function, that first piece is going to be zero. Um, whereas, you know, as I go to this 14 is still four. But then times the drift of a natural law Globex. That's one over X and same thing with this 1/5. That's still there. Times the drift of of natural log of a function is one over that function. Yeah, but then you have to take the derivative of the inside. This is the chain rule, which is four X And you know, some teachers will let you leave your answer like that. I would just clean it up a little bit. Just write us four over X plus four x over five times the quantity of two x squared plus nine. But there's really no reason to simplify any more than that. That should be good enough. Um, you could try to get this as a single denominator, but I don't see the reason this has to do that. This should be good enough

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