Using differentials, find the approximate values of the following:

Let us assume that

Also, let x = 0.09 so that x + Δx = 0.082

⇒ 0.09 + Δx = 0.082

∴ Δx = –0.008

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

We know

When x = 0.09, 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 = –0.008

⇒ Δf = (1.6667)(–0.008)

∴ Δf = –0.013334

Now, we have f(0.082) = f(0.09) + Δf

⇒ f(0.082) = 0.3 – 0.013334

∴ f(0.082) = 0.286666

Thus,