Using differentials, find the approximate values of the following:

(80)^1/4

Let us assume that

Also, let x = 81 so that x + Δx = 80

⇒ 81 + Δx = 80

∴ Δx = –1

On differentiating f(x) with respect to x, we get

We know

When x = 81, we have

Recall that if y = f(x) and Δx is a small increment in x, then the corresponding increment in y, Δy = f(x + Δx) – f(x), is approximately given as

Here, and Δx = –1

⇒ Δf = (0.00926)(–1)

∴ Δf = –0.00926

Now, we have f(80) = f(81) + Δf

⇒ f(80) = 3 – 0.00926

∴ f(80) = 2.99074

Thus, (80)^1/4 ≈ 2.99074