It’s pretty common to hear that dy/dx isn’t a fraction, but if that’s the case then why do we treat as such in a differential equation or an integral? For example, if dy/dx = f(x), then how can we just write it as dy= f(x)dx as though it’s a fraction?
In: Mathematics
They aren’t fractions, but because of the way they are defined they can often be manipulated as if they are. But keep in the back of your mind, that whenever it looks like they are being manipulated like fractions, there is actually something else going on.
For your example let’s try to solve for y:
dy/dx = f(x);
You ‘multiply’ by dx:
dy = f(x)dx
And then you kind of hand wave-y integrate both sides (even though its dy on one side and dx on the other):
∫ dy = ∫ f(x) dx
y = ∫ f(x) dx
But you can also just think of taking the integral of both sides with respect to x:
∫ (dy/dx) dx = ∫ f(x) dx
On the left side you take the integral of the derivative, which is just the original function y (by the fundamental theorem of calculus). On the right side you just have the integral:
y = ∫ f(x) dx
So you get the same answer.
All manipulations of dy/dx like a fraction can be properly reinterpreted without the fraction manipulation.
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