Don’t be so fast to shoot ChatGPT down or LLMs for that matter.
Prompt
Create the rules of calculus derivatives for Prolog then show step by step how to calculate the derivative of 2(2/t)
Completion
Creating a Prolog program to handle calculus derivatives involves defining rules that capture the basic principles of differentiation. In Prolog, we can define rules for the derivative of constants, power functions, addition, subtraction, multiplication, division, and more complex functions as needed.
For the task at hand, let’s focus on the necessary rules to differentiate the function 2  (2/t)
. We need rules for constants, subtraction, and division. Here’s a simplified version:
 Derivative of a Constant: The derivative of a constant is zero.
 Derivative of a Variable (like t): The derivative of
t
with respect to t
is 1.
 Derivative of Subtraction: The derivative of
f  g
is the derivative of f
minus the derivative of g
.
 Derivative of Division: The derivative of
u/v
is (v * du/dt  u * dv/dt) / v^2
, where du/dt
is the derivative of u
with respect to t
, and dv/dt
is the derivative of v
with respect to t
.
Let’s define these rules in Prolog:
% Derivative of a constant is 0
derivative(C, t, 0) : number(C).
% Derivative of t with respect to t is 1
derivative(t, t, 1).
% Derivative of subtraction
derivative(U  V, t, DU  DV) :
derivative(U, t, DU),
derivative(V, t, DV).
% Derivative of division
derivative(U / V, t, (V * DU  U * DV) / (V * V)) :
derivative(U, t, DU),
derivative(V, t, DV).
Now, let’s calculate the derivative of 2  (2/t)
step by step:

Apply the Subtraction Rule:
f = 2
and g = 2/t
df/dt = derivative of 2
and dg/dt = derivative of 2/t

Calculate df/dt
:
2
is a constant, so its derivative is 0
.

Calculate dg/dt
:
 Apply the Division Rule to
2/t
:
u = 2
(constant) and v = t
du/dt = derivative of 2
(which is 0
)
dv/dt = derivative of t
(which is 1
)
 So,
dg/dt = (t * 0  2 * 1) / t^2 = 2 / t^2
.

Combine the Results:
 The derivative of
2  (2/t)
is 0  (2 / t^2)
, which simplifies to 2 / t^2
.
In Prolog, you would query this as:
? derivative(2  (2 / t), t, Result).
And the expected output should be:
Result = 2 / t^2.
This Prolog program is a basic implementation and works for the given function. For more complex functions, additional rules (like for exponentiation, trigonometric functions, etc.) would need to be added.
To cross check using Wolfram
Related post
Define a Prolog predicate that calculates the derivative of a mathematical expression