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

$$\frac{-\left(12 x^{2}+25\right)}{x\left(3 x^{2}+5\right)}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

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Use the properties of loga…

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Use logarithmic differenti…

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Find the dervivative of y=…

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we need to find the derivative of this function the square root of three X squared plus five all over for X to the fifth. Now, the way I look at this problem, I think it is beneficial to rewrite it as natural log. Um, now I'm gonna not right the square root of three X squared plus five. I would write it as three X plus five to the one half power. I think all calculus students should know the square root. No one have power the same. Now, I'm also going to rewrite the denominator because it's dividing. We can split it using subtraction. So natural log a four and minus natural log of X to the fifth. The reason why I would do that is I'm thinking the natural law before when I find the derivative of that that's going to be zero, because natural law before is always the same number. And then I can also move these powers in front using the law of logs. Uh, so what I'm saying is, uh, we have one half natural log of three X squared plus five. We still have natural law before here, as I'm writing it But when I get to the derivative, it will be zero eso from here. When I take the derivative, I'm just thinking about you know how easy the problem is Because one half is a constant. The derivative, a natural log is one over that function three X squared, plus five times the derivative of the inside. Because this is the chain rule, which is six X. I already mentioned the derivative of this zero on. Then this last piece would be five times one of her ex. If you wanted to clean this up, you probably don't even have to. But you can clean it up because half of sixes three. So I'm looking at three X over three X squared plus five a minus five over X, which you could simplify that I just don't really see the benefit of simplifying this other than to get, you know, the same denominator. But that's just busy work at this point. So this is good enough. Thank mhm

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