Using differentials, find the approximate values of the following:

(15)^1/4

Let us assume that

Also, let x = 16 so that x + Δx = 15

⇒ 16 + Δx = 15

∴ Δx = –1

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

We know

When x = 16, 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 = (0.03125)(–1)

∴ Δf = –0.03125

Now, we have f(15) = f(16) + Δf

⇒ f(15) = 2 – 0.03125

∴ f(15) = 1.96875

Thus, (15)^1/4 ≈ 1.96875