By differential coefficient, we mean the derivative of a continuous function. This represents the limit of the ratio of increment in the dependent variable; say ϒ corresponding to a small increment in the independent variable say, x as the latter tends to zero.
To make the point more clear, let y = f(x), Let us suppose that when x is increased by a small increment δx, then ϒ increase by a small increment δY. In such a case, we have
Y + δy = f(x + δx)
Thus, the change in the value of the function is,
(Y + δY) – Y = f(x+δx ) – f(x)
= δY = f(x +x ) – f(x)
= δY/δx = f(x +h)-f(x)/h
Where δY/δx is the incremental ratio of the dependent variable y with respect to x, the independent variable. When δx→ 0, and the limiting value of f(f(x+δx)-f(x)/δx i.e. δY/δx exists, then we say that y is differentiable with respect of y with respect to x.
However, as a matter of convenience if is represented by