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Quadiili 12V 6e iruna dnn (feara dhbanincite (Clas hiau;flx,v) dydx:6 Lux) = drdv0 0 f(x) axay04 Kx,) dxdy5 4 flx) dxdy...

Question

Quadiili 12V 6e iruna dnn (feara dhbanincite (Clas hiau;flx,v) dydx:6 Lux) = drdv0 0 f(x) axay04 Kx,) dxdy5 4 flx) dxdy

Quadiili 12 V 6e iruna dnn (feara dhbanincite (Clas hiau; flx,v) dydx: 6 Lux) = drdv 0 0 f(x) axay 04 Kx,) dxdy 5 4 flx) dxdy



Answers

Find the derivatives of the given functions. $$y=4^{6 x}$$

We want to find the derivative of F. Of X equals one minus six x rays and power of four. At this point in single variable calculus, we should have learned a few shortcuts which make it easier to take derivatives like this without using the traditional definition of the derivative. So let's relate those definitions now they're one through three here where one is the power rule D D. S. X. To the A L A X. To the a minus one to the product rule D D. S. F G equals DFB X. G plus FB GDX and three. The chain rule D. D. S F of G of X is F prime G of X times G prime X. So we can apply the chain rule and the power rule three and one. Just all this easily. We don't need to rewrite our function because it's already in the form that makes it easy to differentiate. So let's differentiate Now F prime is four times one minus six X cube. This is by the power rule. Multiply Enemy multiplied by negative six. Because of the chain rule, the derivative of one minus six X. Inside parentheses negative six. That's what obtained. Final solution. F prime is negative 24 times one minus six execute.

To solve this problem, you have to understand the general of the differences way have to deliver about apartments. Let's expect 40. They use computers to anybody but the aquatics. It's a spectacle extending, and we expect Okay, No recon. And we can rest on the product off the differences. I suppose you have to have the very best above board October coupons. You and the enemy. Can I get a bit about the with respect, X plus, then you can be anybody but you. The suspect X Okay, Let's see how I would describe it. So, what do you have on day Precip? Derivative of the even hunting. So we can take the day with respect to X off the given counsel Delay bx All right, four bike apart for bless Dubai DX TV guru about off X do a. D s six about off white. Let's debate d x I No, here's a coupon, sir. Ex, anyway, So you can apply here for Dr Rule of the differences. Get the consent of way with respect to eggs so we can lie here. General Arda didn't even book a so we can die about up. It's good for the power of X be way PX Why you bless like you eBay b X escape are for Plus we can't take often. Picture outside the rivet, so you can. I poured into eBay, DEA exc about tea. Will do six in court. Delay delay. Like about you may do into do you a great P s you know, derivative of the question. No way we can. Now we can apply the liberty form with the power from allowed that delivered. Suppose you we have to find it. The debate about your dear extra power. And so we can id the enemy unit about up in minus in Minnesota. The boat up X get applied this formula here for, like you for extra power for exponent off extra toe exponent. Why liberty? So we can act like here last week and I exponent export. Do you obey? He s so he can apply here. Duty exponent of way. Do you ever Yes. No. Next way. Then bless like you in Goa. Audience opponent off x TV. You gonna play here? Bless for in tow. Anybody do, it's gonna end up eggs one day. Do each will go. Six. Enough light here six. Do you do exponent of by one They too into do you like right? Deals? Okay. No weekend Simply pay so we can get TV. Exponent of export Exponent of why Oh you away e x less poured into exponent over x TV Let's put it up White TV Bless Begin gets allowed Who? Good days. So we gotta get city six Group X Because we cannot explain End up one day exponent of one day too Look X So you can also cancel our tee times t night You do group way do you a B s Now we can take one side all come up you away dxe so you can get it's It's morning poor Why exponent do do a bay d X minus nine The blue clay You a way Yes, they all left I can't say so. You can die for ex explain on TV Why explaining? T minus six in tow Route eggs Now you can take on one from in this group You have a e x Then we can get Tootie X. Why explain and coo? Then we can get minus minus nine. Why? Into the way? Yes, very good night minus 40. Excuse like you minus four x Q Like you minus six. Look, except now we can divide both sides by t x explanation for why exponent to minus nine or so we can get you ever dx each particle minus four x Q like you minus six eggs divided right t explanation for why it's where minus nine. What began as a guide in the more safety reason again, Dick minus negative. One common from that. This Oh my God. So you can write for X Cube like you minus six x plus. Now we're gonna get my negative on the denominator so you can die minus nine. Qy my TV excess one and four. Why you sweat so you can say hello. You're the one after that. You can get for excuse like you blast six root ex, do you? I didn't write nine y mindless TV ex explanation for why you swing that. Thank you

Okay, so we'll start by taking deserve a tive of both sides. So we have d over DX and we'll start with her left hand side. We have this term here to have an accident. Why? So we're gonna use our product rule. They're starting with extra par four. That gives us four x Cute times are remaining, which is why Cubes, plus the derivative of Y Cube. So that's three y squared times the derivative over inside that b y over the X end, our remaining, which is extinct before okay. And now moving on to our next time, we'll take the derivative of that. Using our power rule. We have four times three open, too. So that's two times three, which is six x to the power of 3/2 minus one. And that's, um, 1/2 which is equal to deter bit of this turnout. So we have six times 3/2, which is equal to nine. And then why'd your power of 1/2 times the derivative of why? So that's d Y over the X and the derivative of a constant That's this sterile. So I'll leave that out. And now I want to move my T Y over TX to our right hand side so we can isolate that so well, with this term to our right hand side. So that leaves us with the following. Okay. And now that we have hors d Y over GX on one side, we can factor that out. So we're left with the fallen. Okay, Now, let's solve for T y over the X who would divide by nine. Why are 1/2 minus three white squared extra power for? Okay, so we get that, um, g y ing over. The X is equal to the fall line.

Okay, We're gonna take the derivative of six X squared plus five y squared equals 36. Implicitly, the derivative of six X squared is 12 eggs call us and then the derivative of five y squared. And that is the derivative of y. Respect to X would be 10. Why times d y the X. And as we've learned earlier, the derivative a constant is just zero. I want to solve this equation for D White yet So I moved the 12 act to the other side, Which gives me Ken Why? And I could write d y. The X is equal to negative 12 x and so now we divide both sides by 10. What? Which tells me that the Y the X is equal to negative 12 x divided by 10. Why? And that will reduce to negative six x over Why? Why? And that's the end


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