# Basic differentiation question.

1. Aug 6, 2012

### 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

2. Aug 6, 2012

### 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.

3. Aug 9, 2012

### 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.

4. Aug 10, 2012

### 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.

5. Aug 11, 2012

### GarageDweller

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