Let f : D → R, D ⊂ R, be a given function and let y = f(x). Let ∆x denote a small increment in x. Recall that the increment in y corresponding to the increment in x, denoted by ∆y, is given by ∆y = f(x + ∆x) – f(x). Fig.
We define the following
(i) The differential of x, denoted by dx, is defined by dx = ∆x.
(ii) The differential of y, denoted by dy, is defined by dy = f′(x) dx or
`dy =((dy)/(dx)) triangle x . `
In case dx = ∆x is relatively small when compared with x, dy is a good approximation of ∆y and we denote it by dy ≈ ∆y.
For geometrical meaning of ∆x, ∆y, dx and dy, one may refer to Fig.
Video link : https://youtu.be/5NxtGK0Bs-U