Some points to know
If a function \(f\relax (x) \) is known at \(n\) points, and we also know that the function is a polynomial of at most \(n-1\) degree, then we can find the polynomial exactly by solving \(n\) equations and finding the \(c_{0},c_{1},\cdots ,c_{n}\) coefficients. Hence no need to do numerical differentiation, we can do analytical differentiation.
Remember this for Taylor: \(f\left (x+h\right ) =f\relax (x) +hf^{\prime }\relax (x) +\frac {h^{2}}{2}f^{\prime \prime }\left ( \xi \right ) \) for this to be valid, \(f\relax (x) ,f^{\prime }\left ( x\right ) \) have to be continuous in the CLOSED interval between \(x\) and \(h\) while \(f^{\prime \prime }\relax (x) \) need to exist on the OPEN interval.
\(f\left (\sqrt {2}+h\right ) -\frac {df}{dx}@\left (\sqrt {2}\right ) \ \)
\(f\left (\sqrt {2}\right ) \ \)
\(f^{\prime }\left (\sqrt {2}\right ) =\frac {f\left (\sqrt {2}+h\right ) -f\left (\sqrt {2}\right ) }{h}\ \)
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