Chain rule Obtain the following derivatives and simplify: f(z) = esin 5 2"(1) f(t) = (tan(1 _ 782))-1 f(z) = sin? (322 f (w) = 3w2 + sin(w? + 6) 622 + 1+ e-2 f...

Derive the chain rule using local linearization. [Hint: In other words, differentiate $f(g(x)),$ using $g(x+h) \approx$ $\left.g(x)+g^{\prime}(x) h \text { and } f(z+k) \approx f(z)+f^{\prime}(z) k .\right]$

Yeah, yeah. So this problem, I'm doing derivatives with product rule and caution rule as with anything in math. If you can simplify earlier is better. Okay, so when I look at this, every term has a Z in it. So factories ation is going to help here so I can write this as F of Z is equal to three Z squared over Z five Z plus seven. That's going to help me because now you can write this as 3Z Over five Z plus seven. Okay, so that means that now, okay, when I do the quotient rule, I'm gonna have lesser degrees of my terms. So that's what I want to work on. So starting from this point, let's look at it and see what is F prime of Z. So F prime of Z. Yeah. Is going to be equal to the derivative of the numerator times the denominator. Yeah minus the numerator times the derivative of the dominator. All of this over the denominator squared. So when you do this, you're going to end up with f prime of Z is equal to 15 Z. Uh huh. Yeah. 15 Z plus 21 minus 15 Z. All of that over five Z plus seven square. Final answer here is 21 Over five Z plus seven quantity squared. Yeah.

So I have a function to your F of omega is six square root of omega plus one over omega square plus five natural log of omega. Were asked to find the derivative. What is helpful? A lot of times when you have powers is to write them in exponential four, six times omega to the 1/2 Plus Omega to the -2 Plus five natural log of mega. Because if you learned the power rule, that's just useful when you have in an exponential form. So now to take the derivative of prime of omega is going to be we'll bring the one half in front. So one half times six. And then omega if you decorate, meant one half by one, so one half minus to have this minus one half power. And then you're gonna have plus -2. Omega to the -3 power plus five. And then the derivative of the natural log is just one over Omega. And so now if we write this we're gonna have f prime of omega is equal to three, so that's three over the square root of omega minus two Over Omega Cube plus five over omega. Uh huh. Mhm.

We're looking for D W D T which first will want to find a W d x times dx DT, which will be negative. Five. Why? Sign X y minus z co Sign of ecstasy Times negative t to the native to And then we're going to be working with our dwt white times. Or do I d t term? Which would be negative. Five x signed X y times one And now for a WDC times d is e d T term. This always gets like a little bit of a mouthful for me. So what a native Exco sign of XY. Times three t squared. And so now let's go ahead and convert and consolidate. And so I was actually a little lazy doing this problem. I didn't actually consolidate the sine x y coastline eggs e in this term. Instead, I kind of kept them out up as kind of like dummy variables. Um, so I kept my sign next. Why here? And it worked out all right just because one of these two terms cancels out, but we'll have our sign of X y and what we'll get from looking at this term here is this is going to turn into a negative five t to the native one, and then we're gonna look for other sign next white terms, which is right here, which is going to be negative five t to the native one. So then this is the term that, of course, cancels out. So that's gonna be equal to zero right there. And then we're gonna add our co sign xz term that we have, which, since I have a little bit foresight here among go ahead and actually make the substitution of the eggs e so we'll have our t squared. And I'm actually gonna put times here just because I wrote the coastline t squared in front, which I don't usually like doing, and then we're gonna be finding it from two terms. We're gonna have a negative Z minus R x there, which will end up giving us e plus three t so we'll have 40 co sign t squared as our final answer

Welcome to New Madrid in the given expression we have to find the derivative of the given parliament. So we didn't right the derivative function on both sides. And then we start thinking what should be our next move. So we have to take the coefficients out and do a town by town different stations, it would be minus six D D X. Of extra power seven plus five into D D X. Of X cube. And then D D X off by square. Now pie is a constant. Therefore we will have zero for this term. So we have minus six into seven. Extra power six minus up seven minus one plus five into three X. Where it should be. Try to be the small step sons. And this would reduce to see. So this is minus 42 extra the power six plus 15 X squared. So if there is a constant term present, that does not change the slope. Okay, how can I explain that? Think of why is equals to a mix and why is equal to a mix plus six? So if you think of bicycles to mx, see any positive in this looks, I just draw a line like this, then this will revise equals two Mx and this will be bias equals Mx plus six. So if you see the slope is not changing and that's how the slope will not, the differentiation will also not change. So I hope you understand that. How irrelevant the presence of a constant is there in this expression? Let me know if you have any questions. Bye bye.