=DERIVXY(x, y, p, [options])
DERIVXY
is a powerful function which employs cubic splines for estimating the derivative at an arbitrary point based on a set of (x,y) data points only.
With options, you can elect to weigh the data points, use exact or smooth least square fit, as well as specify end points slopes if known.
x
a vector of the points xcoordinates.
y
corresponding vector of the points yvalues.
p
the point at which to compute the derivative.
n
order of the derivative. Enter a value of 1, 2, or 3. Default is 1.
ctrl
a set of key/value pairs for algorithmic control as detailed below.
Description of key/value pairs for algorithmic control
Keys  ISLOPE, ESLOPE 
Admissible Values  real number 
Default Value  Unconstrained 
Remarks 

Key  PERIODIC 
Admissible Values (Boolean)  True or False 
Default Value  False 
Remarks 

Key  SFACTOR 
Admissible Values (real)  ≥ 0 
Default Value  0 
Remarks 

w
strictlypositive corresponding set of weights for the (x,y) data points. Default value is unity.
In this example we sample the function
$f\left(x\right)=xsin\left({x}^{2}\right)+1$
then compute its derivative from the sampled data points using DERIVXY
and compare the result to the analytic derivatives given by
${f}^{\text{'}}\left(x\right)=sin\left({x}^{2}\right)+2{x}^{2}cos\left({x}^{2}\right)$
Using the formulas shown in Table 1, we generate by AutoFill the (x,y) data values and the analytic derivatives at the xvalues. The numerical values are shown in Table 2 below:
A  B  C  
3  x_data  y_data  Analytic derivatives at x values 
4  0  =A4*SIN(A4^2)+1  =SIN(A4^2)+2*A4^2*COS(A4^2) 
5  0.25  =A5*SIN(A5^2)+1  =SIN(A5^2)+2*A5^2*COS(A5^2) 
6  ⇓ Drag down to row 20 
A  B  C  
3  x_data  y_data  Analytic der 
4  0  1  0 
5  0.25  1.015615  0.1872153 
6  0.5  1.123702  0.7318602 
7  0.75  1.399977  1.4849677 
8  1  1.841471  1.9220756 
9  1.25  2.249957  1.0258913 
10  1.5  2.16711  2.0487081 
11  1.75  1.138268  6.0268418 
12  2  0.5136  5.9859515 
13  2.25  1.1135  2.5335616 
14  2.5  0.917052  12.459939 
15  2.75  3.634003  5.3043158 
16  3  2.236355  15.988226 
17  3.25  1.94996  9.773066 
18  3.5  0.08892  22.972966 
19  3.75  4.739551  3.0952707 
20  4  0.15161  30.933007 
Using DERIVXY
formulas shown in Table 3, we generate by AutoFill the numerical derivatives at the xvalues and compute the relative error with respect to
the analytical derivatives. The numerical results are shown in Table 4 below.
Note that we have passed defined names
x_data and y_data for columns A4:A20 and B4:B20 respectively. This is one way to lock the data so AutoFill
ignores these arguments and increments only the third argument. Alternatively we could use a dollar sign, $, to lock the first two arguments during AutoFill using the
formula =DERIVXY($A$4:$A$20, $B$4:$B$20, A4)
.
E  F  
3  Numerical Derivatives  % Errors 
4  =DERIVXY(x_data,y_data,A4)  1 
5  =DERIVXY(x_data,y_data,A5)  =ABS((E5C5)/C5) 
⇓ Drag down to row 20 
E  F  
3  Numerical Derivatives  % Errors 
4  0.006256108  100.00% 
5  0.189289352  1.11% 
6  0.733522457  0.23% 
7  1.488966928  0.27% 
8  1.923837894  0.09% 
9  1.015441218  1.02% 
10  2.07793704  1.43% 
11  6.043962293  0.28% 
12  5.914791221  1.19% 
13  2.68187738  5.85% 
14  12.35516509  0.84% 
15  4.867529358  8.23% 
16  15.99364056  0.03% 
17  7.900514125  19.16% 
18  19.69241868  14.28% 
19  9.404971886  203.85% 
20  58.06465252  87.71% 
Notice that the errors are largest near the end points of the data since the default spline fit
does not constrain the derivatives at the end point. But in this example we know the exact
derivatives at the end points from the analytical formula. We can improve the accuracy by supplying
this data using the keys ISLOPE and
ESLOPE in the optional control
argument for DERIVXY
.
We name the range K1:L2 end_slopes and define the following Key/Value pairs for the start and end slope
using the computed analytical derivative values in C4 and C20 from Table 2 above.
K  L  
1  ISLOPE  0 
2  ESLOPE  30.933 
We regenerate the new derivatives and errors in columns H4:H20 and I4:I20 from the enhanced DERIVXY formulas in Table 5. The numerical values obtained by AutoFill are shown in Table 6 below.
H  I  
3  Nummerical Derivative with end slopes  % Errors 
4  =DERIVXY(x_data,y_data,A4,1,end_slopes)  1 
5  =DERIVXY(x_data,y_data,A5,1,end_slopes)  =ABS((H5C5)/C5) 
⇓ Drag down to row 20 
H  I  
3  Num. Der. with end slopes  % Errors 
4  0  0.00% 
5  0.187612966  0.21% 
6  0.733971891  0.29% 
7  1.488845578  0.26% 
8  1.923873859  0.09% 
9  1.015418708  1.02% 
10  2.077882964  1.42% 
11  6.044156087  0.29% 
12  5.91407012  1.20% 
13  2.679186771  5.75% 
14  12.36520643  0.76% 
15  4.830054635  8.94% 
16  15.853783  0.84% 
17  8.422469647  13.82% 
18  21.64038321  5.80% 
19  2.135069296  31.02% 
20  30.93300669  0.00% 
Notice the improvement in the relative error with the defined end points slopes. Keep in mind that we are working with a relatively small sample size. To obtain any further improvement would require finer sampling.
DERIVXY
computes the derivative by fitting a cubic spline model to the data.