## Task Condition

Approximately calculate using the differential.

$y=x^{7}, x=2,002$
## Solution

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

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

Choose:

$x_{0} = 2$

Then:

$\Delta x = 0.002$

Calculate:
$y(2) = 2^{7} = 128$

$y^{\prime} = \left(x^{7}\right)^{\prime} = 7 x^{6}$

$y^{\prime}(2) = 7 \cdot 2^{6} = 448$

We get:

$y(2.002) \approx y(2) + y^{\prime}(2) \cdot 0.002 = 128 + 448 \cdot 0.002 = 128.896$

## Problem Kuznetsov Differentiation $5-21$