## Task Condition

Approximately calculate using the differential.

$y=x^{6}, x=2.01$
## 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:

$x_{0} = 2$

Then:

$\Delta x = 0.01$

Calculate:

$y(2) = 2^{6} = 64$

$y^{\prime} = \left(x^{6}\right)^{\prime} = 6 x^{5}$

$y^{\prime}(2) = 6 \cdot 2^{5} = 192$

We get:

$y(2.01) \approx y(2) + y^{\prime}(2) \cdot 0.01 = 64 + 192 \cdot 0.01 = 65.92$

Problem Kuznetsov Differentiation 5-13