Question 652639
<pre>
We see that 399 is 1 less than 400, whose square root is 20 

We start with the square root function:

 y = <font face="symbol">Ö</font><span style="text-decoration: overline">39</span>

 y = {{{ matrix(2,1,"",x^(1/2))}}}

dy = {{{1/2}}}{{{ matrix(2,1,"",x^(-1/2))}}}dx

dy = {{{1/2}}}{{{ matrix(2,1,"",x^(-1/2))}}}dx

dy = {{{1/2}}}{{{ matrix(2,1,"",1/x^(1/2))}}}dx

dy = {{{1/2}}}{{{1/sqrt(x)}}}dx

dy = {{{1/(2sqrt(x))}}}dx

We let x = 400.  The change in x = 400 to 399 is -1, so we let dx = -1

dy = {{{1/(2sqrt(400))}}}(-1) = {{{-1/(2*20)}}} = {{{-1/40}}}

dy = -.025

So we add -.025 to 20

20 - .025 = 19.975

That's the approximate square root of 399.

Checking with a calculator, <font face="symbol">Ö</font><span style="text-decoration: overline">399</span> = 19.97498436, very close.  

Edwin</pre>