Discrete derivative representation

Question

I have a strange question about the representation of discrete function derivative. Suppose I have the following $f(x)$ function. If I compute its derivative, it gives me this figure A. However, this representation is for me counter intuitive, I'll be rather using this representation figure B. This latter representation (figure B) gives me nice areas where I can clearly see the integral... Thus, I was wondering if I should use the first or the second representation as the discrete derivative of $f(x)$? I agree, it might be a strange question but this problem confuse me a lot since some people define for a given time serie that $\dot{x}=\frac{x_t-x_{t-1}}{\Delta t}$.