What Does Δy/Δx Represent in Basic Differentiation?

[raeshun]

what exactly does Δy/Δx mean.
for instance i know that when
y=x²
Δy/Δx=2x
but what does Δx/Δy equal?
also why is the derivative always Δx/Δy?
also what does Δx by itself mean for instance if y=x² what is Δx

i appreciate any and all answers thanks

 

[brhmechanic]

On the graph of a function, such as y=x^2, if you want to find the average slope between two points, you basically choose 2 points and use the basic slope formula to find the secant line. Well, if those 2 points are an infinitely small distance from each other, it forms a tangent line, which is denoted by "dy/dx" or "f'(x)", and gives you the value of the instantaneous rate change for one variable (y), with respect to a different variable (x). But the two variables have to be related or else the answer is just zero.

For your second question, dx/dy is just differentiating x with respect to y (just like in the above paragraph, but differentiating x with respect to y this time). However, anitderivatives and integrals for a function tell you the area bounded by the function and the x-axis between 2 points, and is denoted with F(x)

Lastly, by itself, dx by itself really doesn't have much meaning. It has to be attached to a derivative or an integral so that it tells you what to do with your equation. Other than that, it's pretty useless.

 

[physeven]

I think brhmechanic answered your first 2 questions but to answer your third one: dx on it's own is called a differential. For example, for f(x) = y, dy = f'(x)*dx, and dy is the differential of y. Realize here the the differential dy is a function of f'(x) and dx, so basically one could say that for a function f(x), the differential of a function df is a function of 2 independent variables, x and Δx, and is written as

df(x,Δx) = f'(x)Δx

It is conventional to write Δx= dx, so that you have df(x) = f'(x) dx. Hope that helps out a bit.

 

[meldraft]

Formally speaking, $\frac{df(x)}{dx}=lim_{Δx->0}\frac{f(x+Δx)-f(x)}{Δx}$.

$Δx=x_2-x_1$, or more generally, $Δx=x_{i+1}-x_i$

When differentiating in calculus, we consider that Δx is always the same (thus we cut the abscissa in equal pieces). At least that's how the Riemann integral works.

 

[GarageDweller]

Post is confusing, we don't use deltas for differentials, we just use dx and dy