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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.$$y=2^{3 x^{2}}$$

$$6 x \ln 2\left(2^{3 x^{2}}\right)$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

Missouri State University

Baylor University

University of Michigan - Ann Arbor

Idaho State University

Lectures

02:07

Use the properties of loga…

01:46

03:11

02:22

02:55

01:59

01:22

02:09

01:18

Use logarithmic differenti…

01:19

02:16

So I'm gonna do some sneaky math to get this this answer. And I'm just going to rewrite the problem as why it was too to the three x squared. And remember this because what I'm gonna do is take the natural log of both sides, which is perfectly fine. But the reason why I'm taking the natural log of both sides s So then the exponents at three x squared can get moved in front. It's just like this s only cross that off now because I've moved it. That is a law of logs on my actually make more sense to rewrite this as three natural log of to times X squared. The reason why I'm saying that is because then when I take the derivative three natural log of two is a constant. It's always the same number. So if I did the derivative of the left side, I'm using implicit differentiation. By the way, I hope you've already learned that. Um and on the right side, all I I believe that three natural go to alone and the derivative of X squared is two x Well, if I multiply the two you know in front, I get six natural log of to, but I need to sell for D Y the X A derivative. So I need to multiply. Oh, I should put parentheses around all this because it looks like natural log of two experts Natural log of two times X and also times why which takes me back to the original statement that why is equal to whoops. I missed it right there is equal to two to the three x squared, so I can rewrite that as two to the three x squared. And don't forget about everything else. Six natural log of to times X times two to the three. X squared is the derivative of this problem on, by the way, multiplication community. So the answer key moves at six in front. But, you know, I think we got the premise down here. This is correct.

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