The dunk already happened as you can see but here’s the link if you wanna go marvel at the real thing: https://twitter.com/renatokara/status/1412484734949675013?s=19
It’s been so long since I took calculus. Can you just cancel the diagonal dx, or is the whole point of the joke that you can’t?
You typically can’t do that if you want to be rigorous in math. It’s more complicated.
For literally every conceivable situation that anyone who isn’t a professional mathematician or physicist would ever encounter, yes you absolutely can treat dy/dx as a fraction.
Because it basically is a fraction, either the limit of a fraction as both parts go to zero, or a fraction of two infinitesimals (numbers between 0 and the smallest or positive or negative real number). A lot of mathematicians get sad when you use infinitesimals but it’s fine.
For literally every conceivable situation that anyone who isn’t a professional mathematician or physicist would ever encounter, yes you absolutely can treat dy/dx as a fraction.
Not really.
df/dx=df/dt.
If you pretend they’re fractions you will find dx/dt=1 which is wrong in general. For instance, let’s say f(x)=3, x(t)=sint+t.
There is a lot of confusion that can be caused in instances like that.
EDIT: I suppose in this case you could say df is 0 so you can’t do that, but there is other confusing stuff that can happen if you don’t pay attention to what the derivatives represent. For instance, you may have df/dx(0)=df/dt(0) in which case it is a really bad idea to treat them as fractions.
The df terms in df/dx and df/dt represent fundamentally different things tho, so you couldn’t just cancel them like that even if you’re thinking of it as a fraction. The df term in df/dt is some function of t (say g(t)dt, if you think of dt as an arbitrarily small incriment in t) and in df/dx it’s some function of x (say h(x)dx)).
This turns df/dx =df/dt into (g(t)dt)/dt) = (h(x)dx)/dx, which reduces to g(t)=h(x), which is fine and doesn’t cause any contradictions.
