Using differentials, find the approximate value of `(1. 999)^5`

Explanation

Let x=2 `[therefore 1-0.001=1.999]` and `Deltax=-0.001`
Let `y=x^(5)`
On differentiating both sides w.r.t x, we get
`(dy)/(dt) = 5x^(4)`
Now, `Deltay=(dy)/(dx). Deltax=5x^(4) xx Deltax`
`=5 xx 2^(4) xx [-0.001]`
`=-80 xx 0.001 = -0.080`
`therefore (1.999)^(5) = y+Deltay`
`2^(5)+(-0.080)`
`=32-0.080=31.920`