Approximations

From the figure, y + Δy = f(x + Δx) and y = f(x).

⇒ Δy = f(x + Δx) - f(x)

Dividing both sides by Dx, we get

Δy/Δx = [f(x + Δx) - f(x)]/Δx.... (1)

Let us suppose that lim_Δx→0 Δy/Δx exists.

⇒ lim_Δx→0 Δy/Δx = lim_Δx→0 [f(x + Δx) - f(x)]/Δx

Using first principles, the right hand side of the equation can be written as dy /dx

⇒ lim_Δx→0 Δy/Δx = f '(x) = dy/dx

⇒ Δy/Δx = dy/dt + ϵ         (e>0 and e is a very small quantity)

⇒ Δy = dy/dt . Δx + ϵΔx As ϵΔx is very small.

Δy = dy/dx . Δx (Approximately) …. (2)

Now, f(x + Δx) = Δy + f(x)

(x + Δx) = dy/dx Δx + f(x)…..from (2)

Note : (i) dy ≃ Dy

         (ii) dx = Dx