If 'x' is a quantity, Δx (change in 'x' - also denoted by δx) is a small quantity when compared to x
and x Δx, x Δx2, x Δx3.....are small quantities (when compared to x and Δx) in the decreasing order of magnitude,
then these quantities are called "Infinitesimals" of order 1, order 2, order 3 and so on.
Let y = f(x) be a function defined on an interval A and x ∈ A.
If Δx is any change in x, then let Δy be the corresponding change in 'y'.
Thus Δy = f(x + Δx) – f(x).
Now is called relative change in 'y'.
If exists, then
Thus = + ε, where ε is a small quantity.
From the above we have Δy = Δx + εΔx
Now εΔx is a very small quantity and hence
Let y = f(x) be a function defined on an interval A.
If Δx is any change in 'x',
then .Δx is called differential of y = f(x).
It is denoted by dy or df.
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