Question

Though I ve looked through several pages documenting various differentiation tools in Julia, I have yet to find the following simple functionality. I want to define a function which is differentiable by hand, e.g, f(x) = 2x^3, then call a symbolic computation function to obtain fprime = derivative(f). This code should produce the same results as fprime(x) = 6x^2. Is there a library function which acts like the made-up function derivative above?

Answer 1

The Symbolics.jl package is capable of what you re looking for, although you need to explicitly declare what your variables are

In [2]: using Symbolics

In [3]: @variables x
Out[3]: 1-element Vector{Num}:
 x

In [4]: D = Differential(x)
Out[4]: (::Differential) (generic function with 2 methods)

In [5]: f(t) = 2*t^3
Out[5]: f (generic function with 1 method)

In [6]: expand_derivatives(D(f(x)))
Out[6]: 6(x^2)

You can read more about it here: https://symbolics.juliasymbolics.org/dev/manual/derivatives/

Answer 2

If you work numerically with symbolic derivatives you can also use code-to-code symbolic differentiation by using Zygote

Assume you have f(x) = 2x^3 than having Zygote loaded you can do

julia> f (5)
150.0

to understand what just happened peek into the compilation process to see that the function f has been actually symbolically differentiated.

julia> @code_llvm f (5)
define double @"julia_#79_991"(i64 signext %0) #0 {
  top:
       %1 = mul i64 %0, 3
       %2 = mul i64 %1, %0
       %3 = sitofp i64 %2 to double
       %4 = fmul double %3, 2.000000e+00
  ret double %4
}

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