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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.$$y=\ln \frac{x^{3}(2 x+1)^{4}}{\sqrt{5 x^{2}+4}}$$

$$\frac{3}{x}+\frac{8}{2 x+1}-\frac{5 x}{5 x^{2}+4}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

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02:27

Use the properties of loga…

02:55

03:11

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01:11

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01:59

02:31

Use logarithmic differenti…

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03:34

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01:18

out of this problem is all about manipulating and using the law of logs. And, uh, if you can do that, then finding the derivative is pretty easy. Two X plus one to the fourth fall over square root a five x squared plus four. Yeah, So while you're rewriting this, it's important to recognize that, um, natural leg of execute because it's multiplication. You can add your next log. In this case, it's natural log, and then with a division, it's minus the natural law. And I would expect all of my calculus students to know that the square root is the same thing as to the one half power eso the next law of logs that you shouldn't uses. The power rule. Bring these powers in front, So this is equal to three natural log of X plus four natural log of two X minus one half natural log of five X squared plus four. Yeah, and now we're ready to take the derivative and we're going to do the chain rule. So the first thing is to recognize that the derivative a match log is one over the function on because the multiplication by three that's just going to go into the numerator and same thing with the next one is the forward goes into the numerator derivative of natural August 2 x plus one. But when you do the change really have to take the drift of up to X plus one, which is to get stuck in London numerator. Same thing with us. We have the one half times 1/5 x squared, plus four times the derivative of the inside, which would be 10 x. So as you simplify that, I won't spend a whole day simplifying, you know? But there's things to clean up like 8/2 X plus one and half of 10 would be five. So we're looking at five x or five x squared plus four and that should be good enough. Answer and move on. Yeah.

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