Automatic differentiation using dual numbers

[Thomas Klemm]

To calculate an extremum we have to find solutions of \(f'(x) = 0\).

We can use the SOLVER with the program f'(x) for the example \(f(x) = x^3 - 2x^2 + 2\):

Code:

00 { 62-Byte Prgm }
01▸LBL "f'(x)"
02 MVAR "x"
03 RCL "x"
04 1
05 COMPLEX
06 STO "x"
07 XEQ "f(x)"
08 COMPLEX
09 STO "x"
10 RTN
11▸LBL "f(x)"
12 RCL "x"
13 2
14 -
15 RCL "x"
16 XEQ "X↑2"
17 XEQ "*"
18 2
19 +
20 END

SOLVER

Select Solve Program
f'(x)

1 x
x=1

2 x
x=2

x
1.333333333333333333333333333333333

---

I must admit that I don't fully understand the second part which involves \(\sqrt[3]{x}\).
As there is no built-in function we'd use:

Code:

RCL "x"
3
1/X
XEQ "Y↑X"

For negative values we'd have to use:

Code:

RCL "x"
+/-
3
1/X
XEQ "Y↑X"
+/-

Do you have a specific example in mind?