Context: My economics professor wrote on the board today that $f(K,L+1)-f(K,L)=MPL=\frac{\partial f}{\partial L}$ with $K$ the capital variable, $L$ the labor variable and MPL the marginal product of labor. I think he was wrong.
Is $f(x,y+1)-f(x,y)= \frac{\partial f}{\partial y}$?
Thought about it after class and came up with the counter example $f(x,y)=x^2+y^2$ but not sure of my answer.
The first (partial) difference $f(x,y+1) - f(x,y)$ is not the first (partial) derivative $\partial f / \partial y$.
At best, one may say that the first difference is a (crude) approximation for the first derivative and $$\frac{f(x,y+1) - f(x,y)}{(y+1) - y} \approx \lim_{\epsilon \rightarrow 0} \frac{f(x,y+\epsilon) - f(x,y)}{(y+\epsilon) - y}$$