Q:

Why does forward-difference give a different value than backward-difference?

We are given that for $x_0 = 0.5, x_1 = 0.6$ and $x_2 = 0.7$, we have $f(x_0) = 0.4794, f(x_1) = 0.5646$ and $f(x_2) = 0.6442$. Also $h = 1$.
The forward-difference formula is $f'(x) = \dfrac{f(x_i + h) - f(x_i)}{h}$ and the backward-difference formula is $f'(x) = \dfrac{f(x_i) - f(x_i - h)}{h}$ . 
Now, using forward-difference for $f'(0.6)$, I get 
$\displaystyle f'(0.6) = \frac{f(0.7) - f(0.6)}{0.1} =  \frac{0.6442 - 0.5646}{0.1} = 0.796$. 
However, when using backward-difference for $f'(0.6)$, I get
$\displaystyle f'(0.6) = \frac{f(0.6) - f(0.5)}{0.1} =  \frac{0.5646 - 0.4794}{0.1} = 0.852$.
Clearly, $0.796$ and $0.852$ are not the same value. 
So, what is the value for $f'(0.6)$?  The answer in the back of the textbook says that it is $0.852$.  Why is it $0.852$ and not $0.796$? 

A:

Each is only an approximate derivative. The textbook happens to assume you use the backward derivative. While neither of these two options has an inherent advantage over the other, $(f(x+h)-f(x-h))/(2h)$ is typically much more accurate than either of them.

