## Task Condition

Approximately calculate using the differential.

$y=x^{5}, x=2,997$
## Solution

If the increment $\Delta x = x - x_{0 \text{ of the argument }} x$ is small in absolute value, then $f(x) = f\left(x_{0} + \Delta x\right) \approx f\left(x_{0}\right) + f^{\prime}\left(x_{0}\right) \cdot \Delta x$

## Let's choose:

$x_{0} = 3$

Then:

$\Delta x = -0.003$

We calculate:

$y(3) = 3^{5} = 243$

$y^{\prime} = \left(x^{5}\right)^{\prime} = 5 x^{4}$

$y^{\prime}(3) = 5 \cdot 3^{4} = 405$

We get:

$y(2.997) \approx y(3) + y^{\prime}(3) \cdot (-0.003) = 243 - 405 \cdot 0.003 = 243 - 1.215 = 241.785$

## Problem Kuznetsov Differentiation 5-24