Slope of a production function Let $F(K,L)$ be a production function with variables $K$ for capital and $L$ for labor.

The slope of the $F(\overline K,L)$ ($K$ taken constant) is defined as the marginal product of labor ($MPL$) such that:

$$MPL=F(K,L+1)-F(K,L)$$

Most production functions have a positive decreasing slope due to diminishing marginal product and therefore are not straight lines. How is the $MPL$ formula still valid? Isn't that the same formula to find a gradient $m$ of a line (i.e. $\frac{y_{2}-y_{1}}{x_{2}-x{1}}$)? If we take any function whose representation isn't a line, its slope at a certain point is equal to the slope of the tangent at the point and not the formula for the gradient $m$.

Also, supposing the given formula for $MPL$ is true, shouldn't we also have $MPL=\frac{\partial F}{\partial L}$?
The (partial) derivative of a continuous function is defined as
\begin{align}
\frac{\partial F(\overline K, L)}{\partial L} := \lim_{\Delta L \to 0}\frac{F(\overline K, L + \Delta L) - F(\overline K, L)}{\Delta L}.
\end{align}
Now if $L \in \mathbb{N}$, then you have a lower bound for the increment $\mathbb{N} \ni \Delta L \geq 1$. Otherwise the definition above is not well defined for $\Delta L = 0$.
Such that we finally arrive at the approximation
 \begin{align}
\frac{\partial F(\overline K, L)}{\partial L} \approx \frac{F(\overline K, L + 1) - F(\overline K, L)}{1}.
\end{align}