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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.$$y=\ln \frac{2 x^{3} \sqrt{x^{2}+4}}{\sqrt[3]{3 x^{2}+7}}$$

$$\frac{3}{x}+\frac{x}{x^{2}+4}-\frac{2 x}{3 x^{2}+7}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

Oregon State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

03:11

Use the properties of loga…

03:52

02:15

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02:07

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06:01

Use logarithmic diff…

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Use logarithmic differenti…

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Yeah, right The way Thio do This derivative is all about separating these pieces. So we have two times x times What? It's execute times the square root of X squared plus four. Yeah, all over the cube root Uh, three x squared plus seven. So the clue in the directions were perfect where they want you to separate things. Eso natural log of two because of the multiplication. Separate with edition natural log of X cubed and other multiplication eso natural log of now I hope everybody in calculus knows that That's the square roots. The same is to the one half power. And then with the vision you want to subtract in division tells you to subtract and add a natural log. And I should have gave myself more room. And just like the square roots to the one half power the cube roots to the one third power. So now when you take the derivative, it's pretty straightforward. My right smaller So I could fit on one screen. Well, the first thing is, this is a constant in the derivative of a constant, always zero. So that's gonna be gone. Um, maybe what I should do is bring a three in front first eso natural of X cubed by the law of logs three Natural log of X. So that's going to be the derivative. Leave the three alone and Europe Natural. Lagerback's is one of our X. Same thing with this next one. You know, you leave that one half alone by doing the law of logs. The derivative natural logs one over X squared, plus four times the derivative of the inside wall. The drift of of X squared is two x and ribbed of a 40 and minus. Same thing with this one third. I wish a fit it all in one line, so it looked prettier. Um, Times 1/3 X squared, plus seven times the derivative of the inside, which is six X. Now, I would clean this up just a little bit. I wouldn't spend all day doing that, like right. It s three over X and see half of two would cancel. So we're looking at plus X over X squared, plus forward. The same thing here is one third of six is two. So that would be minus two X over three X squared plus seven on. That's your answer

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