The exercise with solution is:
$$$$
The way I tried to solve it is:
$x^2 + y^2 = 15^2$ $$$$ $dx = t\frac{dx}{dt} = 12\frac{1}{4} =3$
$x_{before} = 10$
$x_{after} = x_{before} - dx = 10 - 3 = 7$ $$$$ $y_{before} = \sqrt{15^2 - x^2_{before}} = \sqrt{15^2 - 10^2} = 11.18$
$y_{after} = \sqrt{15^2 - x^2_{after}} = \sqrt{15^2 - 7^2} = 13.27$
$dy = y_{after} - y_{before} = 13.27 - 11.18 = 2.09$ $$$$ $dy/dt = \frac{2.09}{12} = 0.174$
compared to the result of the original example: $y' = \frac{7}{4\sqrt{176}} = 0.134$ $$$$ What is wrong with my method ? or is it just the margin of error of the derivatives (if so then why it is used in this case?)