## Task Condition

Approximately calculate using the differential.

$y=\arcsin x, x=0.08$
## 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:

$x_{0} = 0$

Then:

$\Delta x = 0.08$

Calculate:

$y(0) = \arcsin 0 = 0$

$y^{\prime} = (\arcsin x)^{\prime} = \frac{1}{\sqrt{1 - x^{2}}}$
$y^{\prime}(0) = \frac{1}{\sqrt{1 - 0^{2}}} = 1$

We get:

$y(0.08) \approx y(0) + y^{\prime}(0) \cdot 0.08 = 0 + 1 \cdot 0.08 = 0.08$

## Problem Kuznetsov Differentiation 5-5