why does dx/dy = 1/dy/dx??

Question

how would i know this this rule applies?

i know it does to many functions. but could someone explain to me why this is?

thank you guys

Answer 1

i'm going to write it a little simpler:

x/y = 1/y/x
y(x/y) = y(1/y/x)
x = 1/(1/x)
x(1/x) = [1/(1/x)] * (1/x)
1 = 1

Answer 2

If y= f(x) has an inverse x = h(y) that is also a function and both are continuous in the same neighborhood, then
h'(y1) = lim delta y -->0 (delta y/delta x)
= lim delta x --> 0 1/(delta y/ delta x)
= 1/(lim delta x --> 0 (delta y/delta x) = 1/f'(x1). This shows that the inverse function h(y) has a derivative at y=y1, which is related to the derivative of f(x) by the relation h'(y) =1/ 1/f'(x).

Since f'(x) = dy/dx and h'(x) = dx/dy, we have he theorem, If y = f(x) is a single-valued function which has a derivative which is not 0 in a given interval, then the inverse function x =h(y) exists and has has a derivative, and dx/dy = 1/dy/dx.

Answer 3

The crucial point, and the reason why dy/dx notation is so widely used is that while dy/dx is not a fraction, it can always be treated as one. The derivative is the limit of a fraction- in proving the "chain rule", dy/dx= dy/dt dt/dx, or the "inverse derivative rule", dx/dy= 1/(dy/dx), we can always go back "before" the limit, use the fraction property, then take the limit again. "Differential notation" was developed spefically to make use of that fact.

Answer 4

Think about it. It is simply the law of fractions and inverse fractions.


2/3 = 1/(3/2)

Answer 5

Well 1/(dy/dx) is the same as 1x(dx/dy), so it always applies.