How do I find the gradient of a noise value which has been re-evaluated on a curve

Chain rule says

$$\frac {d} {d x}f(g(x)) = f^\prime(g(x)) \cdot g^\prime(x) $$

In your case you have three functions to compose:

• Your filter function \$f(i)\$ that you approximate stepwise
  (You may get better results if you use a piecewise linear or spline interpolation here, to avoid stairstep artifacts from the truncation. Below, we'll pretend it's a smooth function so the flat steps don't zero-out the gradient)

• Your range remapping function \$r(i) = (\text{filter length} - 1)i\$

• Your noise function \$n(x, y, z)\$

So we need to multiply three coefficients here:

$$\frac {\delta} {\delta x}f(r(n(x,y,z))) = f^\prime(r(n(x,y,z))) \cdot r^\prime(n(x, y, z)) \cdot \frac {\delta n(x,y,z)} {\delta x}$$

In code that could look like this:

private void Evaluate(ref float noise, ref float3 gradient, in UnsafeList<float> filter)
{
    int index = (int)(noise * (filter.Length - 1));

    float filtered = filter[index];

    int maxIndex = Mathf.Min(index + 1, filter.Length - 1);
    int minIndex = Mathf.Max(index - 1, 0);

    float approximateSlope = (filter[maxIndex] - filter[minIndex]) 
                             / (maxIndex - minIndex);

    //               f'                   r'                n'
    gradient = approximateSlope * (filter.Length - 1) * gradient;
    noise = filtered;
}