Numerical differentiaon  Example
Example [1]
Find у ’ (1) and у ’’ (1) for function у (х), given tabular:
Values of function у (х)
x 
y 
Δy 
Δ^{2}y 
Δ^{3}y 
Δ^{4}y 
0.96 
0.7825361 






– 86029 



0.98 
0.7739332 

– 1326 




– 87355 

25 

1.00 
0.7651977 

– 1301 

1 


– 88656 

26 

1.02 
0.7563321 

– 1275 




– 89931 



1.04 
0.7473390 




S o l u t i o n . According to the table we’ll receive: h = 0.02, х = 1, х_{0} = 1, hence, q = 0. Having made final differences of function у (х) and using underlined values, from formula (9) we’ll receive:
Similarly, using blacked out values from formula (10) we’ll receive:
For comparison with exact values we’ll notice, that tabulated function is Bessel’s functionat which the first derivative at х = 1 is equal to – 0.4400506, and the second, accordingly, – 0.325147. From here it is visible, that numerical finding of the second derivative gives greater error, than for the first derivative.
