Suppose we estimate the derivative of a function f at a point x $=$ a in the computer using$$a difference quotient$$f$($a$)$f$($a $+$ h$)$ f$($a$)$h$.$There are three kinds of errors that are introduced as a result.$$ A discretization error because f$($x$)$ is not exactly equal to f$($a$+$h$)$f$($a$)$h $.$ Rounding errors in the calculation of f$($a $+$ h$)$ and f$($a$).$ Rounding errors when computing f$($a $+$ h$)$ f$($a$)$ and when computing f$($a$+$h$)$f$($a$)$h $.$The computed value of the difference quotient is then$($fc$($a $+$ h$)$algorithm fc$($a$))-$:algorithm h$$where fc is the computed value of f $($and is$,$ therefore, necessarily a floating point number$).$Let$$s assume that h is a floating point number. Notice, the absolute error between the true$$value of f$($a$)$ and its computed value can be bounded by$|$f$($a$)($fc$($a $+$ h$)$algorithm fc$($a$))-$:algorithm h$|<=$f$($a$)$f$($a $+$ h$)$ f$($a$)$h$+$f$($a $+$ h$)$ f$($a$)$h$$fc$($a $+$ h$)$ fc$($a$)$h$+$fc$($a $+$ h$)$ fc$($a$)$h$($fc$($a $+$ h$)$algorithm fc$($a$))-$:algorithm h$$We have already seen that the discretization error $$f$($a$)$f$($a $+$ h$)$ f$($a$)$h$<=$Mh$2$where M is a bound on $|$f$|.$ Notice, furthermore, that, although we have written an inequality,$$the actual error is actually approximately equal to the right$-$hand side when h is$$small and M is replaced by $|$f$($a$)|.$