Scripting Guide - SAS

Chapter 8 Programming Methods 225
Additional Numeric Operators
expressions for the derivatives. The expression is threaded by inserting assignments to temporary variables of
expressions that are needed in several places for the derivatives.
Here is an example involving an expression involving three variables. Listing all three variables returns the
first derivatives with respect to each. The result is a list with the original expression and then the derivatives
in the order requested. However, note here that JMP creates a temporary variable T#1 for storing the
subexpression x^2, and then uses that subexpression subsequently to save calculations.
result2=derivative(3*y*x^2+z^3, {x,y,z}); show(result2);
result2 = {3 * y * (T#1 = x ^ 2) + z ^ 3, 6 * x * y, 3 * T#1, 3 * z ^ 2}
To take second derivatives, specify the variable as a third argument. Both the second and third arguments
must be lists. JMP returns a list with the original expression, the first derivative(s), and then the second
derivative(s) in the order requested.
second=derivative(3*y*x^2,{x},{x}); show(second);
second = {3 * y * x ^ 2, 6 * x * y, 6 * y}
second=derivative(3*y*x^2,{y},{y}); show(second);
second = {3 * y * (T#1 = x ^ 2), 3 * T#1, 0}
second=derivative(3*y*x^2,{y},{x}); show(second);
second = {3 * y * (T#2 = x ^ 2), 3 * T#2, 6 * x}
NumDeriv takes the first numeric derivative of an operator or function with respect to the value of the first
argument by calculating the function at that value and at that value plus a small delta ( Δ ) and dividing the
difference by the delta. NumDeriv2 computes the second numeric derivative in a similar fashion. These are
used internally for nonlinear modeling but are not frequently useful in JSL. Note that these functions do
not differentiate using a variable, but only with respect to arguments to a function. In order to differentiate
with respect to x, you have to make x one of the immediate arguments, not a symbol buried deep into the
expression.
Suppose to differentiate y =3x 2 at the value of x =3. The incorrect way would be to submit
x=3;
n=NumDeriv(3*x^2);
The correct way is to make x an argument in the function.
x=3;
f=function({x}, 3*x^2);
n=NumDeriv(f(x), 1);
Consider both the mathematical notation and the JSL equivalent for another example:
For fx () x 2
d 2 ( x + Δ) 2 – x 2
d 2
= , it calculates x = ------------------------------ . At x , .
dx Δ
0
= 3 x = 6.00001
dx
x=3; y=numderiv(x^2); // or equivalently: y = numDeriv(3^2);
6.0000099999513
And here are a few more examples:
x = numderiv(sqrt(7));