H prime of three is going to be equal to one over G prime of H of three, which I'm guessing that But we can use this now if we want to figure out You can derive this from just what the definition of Retain in my long-term memory, but I did retain that They might encourage you to memorize this and maybe for the sake of doing this exercise on Khan Academy you might want to memorize it, but I'll tell you 20 years after I took, almost 25 years after I took calculus, this is not something that I Prime of X is equal to, is equal to one over G prime of H of X. To figure out H prime of X or we could just rewrite it this way. G prime of H of three is and so we should be able We can figure out what H of three is and then we can use that to figure out what G prime of whatever With respect to X of X? Well that's just going to be equal to one. ![]() Rule right over there and then that would be equal to, what's the derivative This would be G prime of H of X, G prime of H of X, times H prime of X. So let's take the derivative with respect to X of both sides, derivative with respect to X, and on the left-hand side well Well now let's take theĭerivative of both sides of this. So if we have G of H of X is equal to X, so we put that H of X back there, which is by definition true So let's start with the expression, well let's start with, It and there's a chance that you might see it Of, but it is interesting so we're going to work through ![]() This isn't the type of problem that you'll see a lot How do we figure this out? Well here we're going toĪctually derive something based on the chain rule and How do we figure it out? They gave us G prime and H and G. They don't even give us H prime of three. Alright, so they want us toĮvaluate H prime of three. I could have just swappedĪre somewhat arbitrary. It multiple different ways, but also G of H of X. You could have started with, well you could have done If someone tells you that GĪnd H are inverse functions, that means that H of G of X is X. So we could also view this as H of G of X and I did all of that so we can really feel good about this idea. Point right over here, we could view it as X, so that is X, but we could also view it as H of G of X. ![]() So that's what the function H would do and so we could view this H would get us back to our original value. G and frankly vice versa, then H could go from that That X to another value, which we would call G of X. Some X right over here, G is going to map from Set as the domain of G, so if you start with Have two sets of numbers, let's say one set right over there, that's another set right over there, and if we view that first Ourselves what it means for them to be inverse functions. We are approaching 3.68,Įven though the value of the function is something different.H be inverse functions. But this would be a reasonable inference. Is what the graph look like once again, we're just Where it's approaching 3.68įrom values less than five and values greater thanįive, but right at five, our value is 6.37. So if this, that's 6.37 then at four, 3.37 is about here and it looks like it's approaching 3.68. So 6.37, but as we approach five, so that's four, actually let That this right over here is 6.37, so that's the value of my function right over there. This is five right over here, At the point five the value of my function is 6.37, so let's say Just substitute five what is g of five? It tells us 6.37, but the limit does not have to be what the actualįunction equals at that point. Most tempting distractor here because if you were to The limit would be 3.68 or a reasonable estimate for And we're approaching 3.68 from values as we approach five from ![]() When we're approachingįrom values less than five. So my most reasonable estimate would be, well it look like So a thousandth below fiveĪnd a thousandth above five we're at 3.68, but then at five all of sudden we're at 6.37. But then if we approach fiveįrom values greater than five. I don't, these are just sample points of this function, we don't know exactly what the function is. But then at five all of a sudden it looks like we're We're only a thousandthĪway, we're at 3.68. Let's see at four is it 3.37,Ĥ.9? It's a little higher. Of x seems to be approaching as x approaches five from Is a reasonable estimate for this limit? Alright now let's work It gives us the x valuesĪs we approach five from values less than fiveĪnd as we approach five from values greater than five it even tells us what g What is a reasonableĮstimate for the limit is x approaches five of g of x? So pause this video, look at this table.
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