Find the equation of the curve that passes through (4,5) if its slope is given by the following equation_2x - 4y(x)...

Find the slope of the curve $y=x^{4}$ at $x=2$.

In this question, we have to find a slope off. The tangent are two points, which, given the slope, is nothing but the value of Irish and those points. So we have the groovy question as why it's where that's exit squared is equal. To wire is to the ball for minus two. Works directors to French in would say, We have to buy wine, actually, plus distribute two weeks confrontational virus to the poor force for y que wire that we're using the culture Andrew. And here we have. All right, so this is the differentiation. We are not supposed to simplify this any further by because it will make it will make much more sense when I will actually pull the values. So let's do that. We have excess minus two on why, as one of its poor piece value, So we have two things. One is too, so we have to buy a rash access to so here we'll have minus for on here buys once of this before. Why dash minus two? This becomes, too. Why racial be equal to minus two. So the value of Irish consult minus one. So now it's quite clear that we're not supposed to waste your time to put the fire dash at one side and trade a question and fired us by. Rash is equal to something because we're eventually the placing, extended by by some constant values. So why not do that at this is stop. Only that will say over time, leg ways we're going to do for the next one as well. So we replace X by minus two on we replace Y bi minus one. So the value of X the same does the value while changing. So if you pull dart, be a getting that this will be minus two y, so it will be minus two wire dash. So there's nothing here who, minus two via dash. Then this is seen. Then this will be minus or four wire dash U minus two. So here we have two via dashes stoop to the value of violation. Solders ordered slope at this point from sodas

Okay, so here is given that the slope. So the slope is do I d axe derivative. So the slow, um, off the curve. Right. E x is, um so you are. The X is well, is negative. X over four. Why? Okay, so we can rewrite the differential equation as well as four. Why de y is equal, Teoh. Negative x dx. Okay, so separate are very moles. Get X rays on one side, wise and the other, um And then we have a differential equation in separate form so that we can integrate both sides to integrate left side and we integrate the right side. Okay, so what we get here while the in agro of four Y de y That's just, um, four y squared over two, which is two. Why squared is equal to, Well, negative X squared. Negative X square over two. And then plus our constant c. Okay, um, So what we get here is where all that d y I'm sorry to Why square plus X squared over two is equal to see right? So we can then replace X with zero. And why, with 1/2 right, given the point that were given him and then south for C. So we get to times Well, 1/2 squared, plus Well, zero squared over to what you just plus zero is equal to see. So we get, um, 1/2 square. Um, times two, which is just 1/2. Right. So 1/2 is he going to see someone? Half is equal to see. Okay, then weaken Substituted 1/2 in for C in two y squared, plus X squared over two equal c. And we get to why square plus X squared over shoe is equal to Well, you could 1/2. Okay, so, um well, this implies that, um, the fractions office supplies that. Let's see Exc Well, but that's a first X squared plus four. Why? Square is equal to one. Cancel therefore the equation of the curve we're looking for. There it is. Right. So we get the prayers of the curve is X squared plus four X is equal to one. This is X where our X squared plus four y square is equal to one. All right,

This question asked us for the equation of the curve that passes through the 0.0 or two and his slope X y is X over. Why? Okay, now what we know is that given the fact that we have a slope at acts over why this means that D y o ver de acts this is slope notation in this problem can be written as equal to X over. Why now? We want all the white terms on the left hand side. So we're just gonna rearrange this and then all the ex terms on the right hand side because now we can We're easily integrate. Increased the expo number one divide by the new exponents. Now, how why Squared is X squared plus two c not substituting X's here, wise to we know we now need to solve for C so we should get C equals and we end up with C equals two. Okay, Now substitute this about this value back into our equation plus two c So, plus two times two when they get why squared is X squared plus four. Now remember, we don't want y squared equals. We want y equals We must take the square root. Now notice how it should be positive or negative or plus or minus because we're doing the square root. However, remember why zero is, too, Which means this has to be positive, which means be just disregard the negative solutions. So this is the answer.

As we look at this implicit function X squared, Y minus X, Y squared. And we want where the derivative is defined. What you want to do is start by doing the product rule. So I'd like to do the derivative of the left side is two X. Leave why alone and then plus now leave X squared alone times and the ripped of why? Which is dy dx. Now we also have the product rule here with a negative in front. So one way of thinking is a derivative of X is one. You leave Y squared alone and instead of plus that you have to distribute that negative to it. So leave the Exelon times in derivative of Y squared which is to Y Dy dx and then the derivative of four would be zero. So what your next thing would be and I think you can do all this at once as you can factor out a dy dx on the two terms the this term and this term and everything else. You can move to the right side by adding y scoring to the right side and subtracting the two X, Y to the right side. So when you solve this multiplication problem, what you want to do is divide it over. So we're looking at why squared minus two xy all over X squared minus two. Xy. And we're answering where the slope of the curve is defined. Well it's undefined if you're defining by zero. So you're going to set the denominator Equal to or not equal to zero would be our limitations. Um So that's all I would do. And I would also factor out and X out of that unequal to zero. So one instance would be when x equals zero because you can't divide by zero there. But then also when X. Is can't be equal to Y. Um You know if if that's more true then you have to our minds to I. Zero and you be the vitamin zero. Again I don't know if they're looking for more ordered perish. Um Because it's looking like it's impossible for extra equal zero in the first place because 0 0 will not equal four. Um But if you had that situation that X equals two I so this one might be an extraneous solution. But you said X not equal to two. Y. You could actually figure out what the X. And the Y value would need to be In order for it to equal four. I don't know if they want you to do all that extra work