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6 points) Differentiate the following function by using Chain Rule. Show all neces -sary steps.flz) = sin? Vr2 +r'...

Question

6 points) Differentiate the following function by using Chain Rule. Show all neces -sary steps.flz) = sin? Vr2 +r'

6 points) Differentiate the following function by using Chain Rule. Show all neces - sary steps. flz) = sin? Vr2 +r'



Answers

Find the derivative of the following functions. $$y=-x^{3} \cos x+3 x^{2} \sin x+6 x \cos x-6 \sin x$$

Uh huh. In this problem we wish to find the derivative of climax using the chain rule and previous differentiation shortcuts. Specifically we want to find the derivative Dy dx or Y equals coastline exposed to sign actually ate. This question is challenging understanding of differentiation in particular is challenging ability to apply the chain will be differentiation to solve. So for like with the view of X. Dy dx DF dx equals D F. D. U. D. Dx. Where dx sign up to access coastline to extremes to as an example. So to solve what we need to do is combine the above with previous differentiation techniques that we already learned specifically the chain rule term here is you cosign exposed to sign X rays to the eighth power. That's the solution is G. F. D. U G. D X. Which is simply found here as eight co sign exposed to sign next to the seventh. This is the FDU Times to cosign X. Medicine X. Which is the U. D. X. Where of course we identified you here as I just mentioned.

In discussion. Where you calling about power Role. You bow and you would do You can decode, you know, and times you bow and minus one times your problem and the general of line on the concerned A sign some a new problem. Can you go to the new prime time schools have a new similarly the consomme the new prime getting close to the new brand minus here attempts This time the new in this question were given The wife told you that cause I'm the six x yes, this side on the X square Totally power off half in this question Here. First we can identify the inside square bracket will be that you just will be the end and found by formula gonna wipe them equal to half We've been down and then we have ghosts under six Thanks plus side of the x square half minutes we're gonna be in which minus half and then by the general. When it attempts goes I'm the six x plus side X square derivative. And now would we be this now the with the coast size six X plus site X square power minus a half times for this one we can things only with a new. So we applied this formula together Miners six ex prime times the side on the six X then Plus, for this one, we can think it's only been a year again a blind this formula. We have the X choir prime attempts to goes on the X Square and finally we get equal to a half go size six x plus site X square minus a half This one way you'll derivative gonna get you the six. So we're gonna minus six size six x plus the rift Illness X square encouraging you to x times goes on X square

For this problem. We're talking the derivative of the sign of the log of eight X to the six I've color coordinated the composition of functions right here so we can use the chain rule. We start with the outer function. So the derivative of the sign is the co sign. And we keep the inside the same times the derivative of the log function, Um, which is gonna be one over eight X to the sixth right there. Times the derivative of eight X to the six, which is 48 x to the fifth right there. We can rewrite this whole thing, then as six over x at times the co sign of the log of eight X to the sixth right there as the derivative. Now, some of you may be wondering why we took the derivative of the log as one over X instead of the derivative of the log of the absolute value of X. And the reason why we conduce that in this case, um, is because in the original function, um, X is a positive number right here because we get X to the sixth. Um, and that can only be a positive thing. So the log of the absolute value would be the same is just the log of, um, the input in this case, so we don't have to worry about that.

In this problem, we're looking to find the derivative of F of X. And to do that, we're actually going to have to use the chain rule twice. So to start out, we'll begin with the big picture here. So that's this export up here. So we pull our six down and then we'll keep everything in the parentheses the same. And then our exponents will subtract one from it. So we get five. And now we have to take the derivative of the inside. So two X to the fifth. We pull down the five, get 10 X. When we subtract one from the experiment, we get four. And now we're going to have to do the chain rule on this next part. So we'll pull down our exponents too, leave the inside the same two minus one is one. So we don't need to write an exponents. And now we have to take the derivative of the inside 04 x minus five. And that just gives us four. So we'll just clean this up of it. All the hard work is done. And at this point, all you have to really do is combined the two and the four, and we'll put an eight out front and then for x minus five, and that's going to be your solution to the.


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