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Let's prove with the derivative of e to the x's, and I think that this is one of the most amazing things, depending on how you view it about either calculus or math or the universe. Well we're essentially going to prove-- I've already told you before that the derivative of e to the x is equal to e to the x, which is amazing. The slope at any point of that line is equal to the x value-- is equal to the function at that point, not the x value. The slope at any point is equal to e. That is mind boggling. And that also means that the second derivative at any point is equal to the function of that value or the third derivative, or the infinite derivative, and that never ceases to amaze me. But anyway back to work. So how are we going to prove this? Well we already proved-- I actually just did it right before starting this video-- that the derivative-- and some people actually call this the definition of e. They go the other way around. They say there is some number for which this is true, and we call that number e. So it could almost be viewed as a little bit circular, but be we said that e is equal to the limit as n approaches infinity of 1 over 1 plus n to the end. And then using this we actually proved that derivative of ln of x is equal to 1/x. The derivative of log base e of x is equal to 1/x. So now that we prove this out, let's use this to prove this. Let me keep switching colors to keep it interesting. Let's take the derivative of ln of e to the x. This is almost trivial. This is equal to the logarithm of a to the b is equal to b times the logarithm of a, so this is equal to the derivative of x ln of e. And this is just saying e to what power is equal to e. Well, to the first power, right? So this just equals the derivative of x, which we have shown as equal to 1. I think we have shown it, hopefully we've shown it. If we haven't, that's actually a very easy one to prove. OK fair enough. We did that. But let's do this another way. Let's use the chain rule. So what doe the chain rule say? If we have f of g of x, where we have one function embedded in another one, the chain rule say we take the derivative of the inside function, so d/dx of e to the x. And then we take the derivative of the outside function or the derivative of the outside function with respect to the inner function. You can almost view it that way. So the derivative ln of e to the x with respect to e to the x. I know that's a little confusing. You could have written a d e to the x down over here, but I think you know the chain rule by now. That is equal to 1 over e to the x. And that just comes from this. But instead of an x, we have e to the x. So this is just a chain rule. Well what else do we know? We know that this is equal to this, and we also know that this is equal to this. So this must be equal to this. So this must be equal to 1. Well let's just multiply both sides of this equation by e to the x. We get on the left hand side, we're just left with this expression. The derivative of e to the x times- we're multiplying both sides by e to the x, times e to the x over e to the x. I just chose to put the e to the x on this term, is equal to e to the x. This is 1. Scratch it out. We're done. That might not have been completely satisfying for you, but it works. The derivative of e to the x is equal to e to the x. I think the school or the nation should take a national holiday or something, and people should just ponder this, because it really is fascinating. But then actually this will lead us to I would say even more dramatic results in the not too far off future. Anyway, I'll see in the next video.