![]() 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. For instance, 2 can be multiplied by x by simply writing 2 x there is no need for a "times" symbol between them.H be inverse functions. Since you're in algebra now, you are familiar with multiplication "by juxtaposition" that is, you know that you can multiply things by simply putting them next to each other.
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