Recursively differentiating basic expression

Question

If I have an expression, such as x * x * x, stored in a data structure like: mult(var x, mult(var x, var x))

And I want to implement a function to recursively differentiate the equation (so to 3 * x * x or x*x + (x + x)*x etc, no simplification required), any suggestions how to being this?

Answer 1

You would find a matching differentiation rule and apply it. For example in this case we have the rule (where A and B stand for whole subexpressions)

diff(mult(A, B)) -> add(mult(diff(A),B), mult(A, diff(B)))

The left hand side of this rule matches the formula when we set

A = var x
B = mult(var x, var x)

So we can apply this rule to the formula and get

diff(mult(var x, mult(var x, var x))) ->
  add(mult(diff(var x),mult(var x, var x)), mult(var x, diff(mult(var x, var x))))

Now do the same recursively for the remaining diff operations.

The other other rule you will need here is:

diff(var x) -> 1