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 limΔx→0 Δy/Δx exists.
⇒ limΔx→0 Δy/Δx = limΔx→0 [f(x + Δx) - f(x)]/Δx
Using first principles, the right hand side of the equation can be written as dy /dx
⇒ limΔx→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
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 limΔx→0 Δy/Δx exists.
⇒ limΔx→0 Δy/Δx = limΔx→0 [f(x + Δx) - f(x)]/Δx
Using first principles, the right hand side of the equation can be written as dy /dx
⇒ limΔx→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