## Task Condition

Approximately calculate using the differential.

$y=x^{21}, x=0.998$
## Solution

If the increment $\Delta x = x - x_{0}$ of the argument $x$ is small in absolute value, then

$f(x) = f\left(x_{0} + \Delta x\right) \approx f\left(x_{0}\right) + f^{\prime}\left(x_{0}\right) \cdot \Delta x$

Choose:

$x_{0} = 1$

Then

$\Delta x = -0.002$

Calculate:

$y(1) = 1^{21} = 1$

$y^{\prime} = \left(x^{21}\right)^{\prime} = 21 x^{20}$

$y^{\prime}(1) = 21 \cdot 1^{20} = 21$

We get:

$y(0.998) \approx y(1) + y^{\prime}(1) \cdot (-0.002) = 1 - 21 \cdot 0.002 = 0.958$

## Problem Kuznetsov Differentiation 2-11