In the textbook Ordinary Differential Equations by Tenenbaum and Pollard, Lesson 45 on improvement of polygonal starting method P641:
Let $y(x)$ be a particular solution:
$y' = f(x,y)$
It says on P642 that "By differentiation of the above formula, we assume the derivatives exist-we obtain"
$y'' = \frac{\partial f(x,y)}{\partial x} + \frac{\partial f(x,y)}{\partial y} \frac{dy}{dx}$
Question is how did the author reach to this formula with $\frac{\partial f(x,y)}{\partial y} \frac{dy}{dx}$?
