basic difference

General

$$f^{\prime }(x)\approx c_{1}f(x+h)+c_{2}f(x)+c_{3}f(x-h)$$

by Taylor expansion

$$f(x+h)=f(x)+h{f}^{\prime }(x)+\frac{h^2}{2!}{f}^{\prime \prime }(x)+\cdots +\frac{h^n}{n!}{f}^{(n)}(x)$$ $$f(x-h)=f(x)-h{f}^{\prime }(x)+\frac{h^2}{2!}{f}^{\prime \prime }(x)-\cdots +\frac{h^n}{n!}{f}^{(n)}(x)$$

Forward difference

$$f^{\prime }(x)\approx \frac{f(x+h)-f(x)}{h}=f^{\prime }(x)+Ch+O(h^2)$$

Center difference

$$f^{\prime }(x)\approx \frac{f(x+h)-f(x-h)}{2h}=f^{\prime }(x)+C{h}^2+O(h^4)$$