Not quite on the caliber of the discovery of noble gas chemistry by preparing a chem 101 lecture (like was done historically) but another illustration why (at least subjectively) teaching elementary courses can still be very rewarding.
Let u = arcsin(exp(x)) du = (1/sqrt(1-exp(2x)) • exp(x) dx We know from trigonometry that, tan(arcsin(β)) = β/sqrt(1 - β^2), which is exactly equal to du/dx when β = e^x I think many people learning calculus would immediately recognize any term resembling x/sqrt(1 - x^2) the second they see it, and substituting u = “the inside” seems to be a fairly natural move as well. I don’t think the integral would be particularly difficult if given to an average calculus student for the first time.
You can also let e^x= sinθ (in order to get arcsin(sinθ) = θ) Therfore, e^x dx= cosθ dθ So our integral ∫ tan(arcsin(e^x)) dx Becomes like this ∫ tanθ cotθ dθ Simplifying this to ∫ dθ = θ + C And since e^x= sinθ So θ in terms of x is gonna be arcsin(e^x)
Given any function g, we want to find a function f for which f(g(x)) is equal to g'(x). In this case, we could just define f(x) to be g'(g^-1(x)), assuming that g is one-to-one (or injective). For example, let g(x)=x^3. Then, f(x)=3(x^(1/3))^2=3x^(2/3). For g(x)=x^2, we must restrict x to nonnegative values (to avoid needing an absolute value), and f(x) would then be 2√x.
So, i just finished rewatching your 100 derivatives, integrals and series vids (took me a while, but i wantend and needed a refresher). With this vid, i had a Leonardo DiCaprio- meme moment where he points to his tv screen after having a major realisation (its from “once upon a time in hollywood” Movie). This is freaking awesome!
Incredible as it is, it's just a magic trick, as there is no actual magic. A covert magic trick is that I can (almost) see it. Thanks, pro bro. You make it look easy.
It was pretty clear if you look at the general formula of the derivative of the inverse function. (d/dx)(f⁻¹(x)) = 1/f'(f⁻¹(x)). Let f(x) be sin(x) and magic is oncoming. This way you can generate your own "f(x)" (as described in this video) when you have your "g(x)" ready.
I'll give two more examples this integral working, besides trig functions: f(y)= e^-y where the integral becomes int(e^(- ln x)) dx f(y) = y^2 where the integral becomes int( (-1/x)^2) dx
I bet people already said it but just as well: this video is another way to look at the inverse function theorem. sec(x) = 1 / (sin(x))', as the theorem states: f^(-1) ' (b) = 1 / f ' (a) where f(a) = b.
Help, I tried to do it with e^(2x) *sen(x) and there's no way to do the inverse or it's derivative, I tried wolframalpha and symbolab and both refused The derivative is 0.2 e^(2x) *(2senx-cosx)
I got a question for part f(y) = d(y)/dx Become 1/f(y) dy = dx Maybe i am wrong or maybe not, just comment if i wrong but my teacher in ODE say that That we can't separate the dy/dx, Because they are one unit, can't be separated. But maybe i wrong tq 🙏😁
A lotta calculus works on the fact that dy/dx can be separated. Like, we are taught that dy/dx is not "dy divided by dx" but we eventually started imagining it to be dy divided by dx to simplify stuff. It isn't wrong at all.
Lmao do you have nothing else to do you're like a 50 year old man still commenting irrelevant mean spirited comments on all of this guys videos, which are aimed at a younger audience to educate will little prerequisite knowledge. Just makes you come off as weird and sad, esp. as you've made so many comments like this, it's pathetic
@@User-he6zd Ditch the silly mic. With my pencils and math knowledge tied behind my back, I can out math this guy, I am a tutor. This guy uses a gimmick and takes the long way to explain things.
@@frankcabanski9409 Jesus christ.... lol he's holding a mic in his hand so his viewer's/students can hear him better. 🤣 your obnoxious for a guy that teaches fucking basic math you can learn at a library for a $1.50
This magic trick looks like a failure of notation. sin^-1 is the inverse of the antiderivative of the reciprocal of sec. So A f(AR(f)^-1) = AR(f)^-1. Differentiate and get f(AR(f)^-1)=D(AR(f)^-1). Which starts looking like algebra, not calculus.
@@ethohalfslab So are a lot of other things; it doesn't make them any less confusing or wrong. Particularly if you want then to indicate f(x)·f(x) as f²(x).
@@dlevi67 are you gonna go yell at a 3rd grader "USING X FOR MULTIPLICATION IS WRONG"? I wouldn't think so, because that notation is fine for their usecase. If this math problem used something like f^2(x) for something, then it's fine to request arcsin and the like. But it doesn't use f^2(x).
@@ethohalfslab Except that: 1. BPRP uses that notation consistently - just as he uses consistently f²(x) instead of the longer (f(x))². 2. The multiplication symbol is _not_ an 'x'; it's '×' - it's a different notation and not an abuse. 3. If you are teaching, you should teach best practice compatible with your (average) student's ability to understand and knowledge. Someone who follows a video like this one _does_ understand and knows what f²(x) means even if it is not used in this specific case.
