Issues with differentiating with respect to polynomial coefficients

[vivekbhattacharya]

I am interested in taking the derivative of a polynomial with respect to its coefficients. When the coefficients are such that all terms of order $n \ge k$ for some $k$ are 0, automatic differentiation packages return 0 as the derivative with respect to those coefficients. Of course, the true derivative of $\beta_0 + \beta_1 x + \cdots + \beta_K x^K$ with respect to $\beta_n$ should be $x^n$.

Here is an example:

using Polynomials
using ForwardDiff

f(x) = Polynomial(x)(0.5)
∇f(x) = ForwardDiff.gradient(f, x)
true_∇f(x) = 0.5.^(0:length(x)-1)

all_params = [[1.0, 2.0, 3.0], [1.0, 0.0, 3.0], [1.0, 0.0, 0.0]]
for param in all_params
    println("Params are $(param).") 
    println("Gradient from ForwardDiff: $(∇f(param))") 
    println("True gradient: $(true_∇f(param))")
    println("")
end

The output from this example is below. Note the issue in the final example.

Params are [1.0, 2.0, 3.0].
Gradient from ForwardDiff: [1.0, 0.5, 0.25]
True gradient: [1.0, 0.5, 0.25]

Params are [1.0, 0.0, 3.0].
Gradient from ForwardDiff: [1.0, 0.5, 0.25]
True gradient: [1.0, 0.5, 0.25]

Params are [1.0, 0.0, 0.0].
Gradient from ForwardDiff: [1.0, 0.0, 0.0]
True gradient: [1.0, 0.5, 0.25]

Is there a way to avoid this issue?