How to compute the derivative of multi octave perlin noise - graphics

Question

I have implemented Perlin Noise (3d) that will not just yield me a field value, but also an analytical derivative. For the analytical derivative, I followed the scratchapixel guide.
This is all fine, but I do not know how to treat the derivative if I have multiple octaves.
For the field value, the octave summing is very straightforward:
Assume I have 4 octaves of noise with persistence 0.5 and lacunarity 2, I would simply do:
float sample_4o( vec3 samplepos )
{
const float amp0 = 1;
const float amp1 = 0.5f;
const float amp2 = 0.25f;
const float amp3 = 0.125f;
const float freq0 = 1;
const float freq1 = 2;
const float freq2 = 4;
const float freq3 = 8;
const float v0 = perlin( freq0 * samplepos );
const float v1 = perlin( freq1 * samplepos );
const float v2 = perlin( freq2 * samplepos );
const float v3 = perlin( freq3 * samplepos );
const float maxval = (amp0+amp1+amp2+amp3);
return ( 1 / maxval ) * ( amp0 * v0 + amp1 * v1 + amp2 * v2 + amp3 * v3 );
}
It's nothing more than a weighted sum of the octaves. Easy peasy.
But now that I have not just four noise values, but also four derivatives (or gradients), how would I combine the four 3d gradients g0,g1,g2,g3 using the amplitudes amp0,amp1,amp2,amp3?
To visualize the example...
I have two signals, the red and the orange.
I know the derivatives for the red and orange signal.
I scale up the freq of orange, to get the dashed-orange signal.
I then scale down the amplitude of dashed-orange, to get the dotted-orange signal.
I add dotted-orange to red, to obtain the 2-octave signal in blue.
How is the derivative of the blue signal expressed in the derivatives of orange and red?

Answer 1

If your gradients are correct, then you just sum them with the same same weights that you use to sum the amplitudes.
Note, however, that gradients are proportional to frequency, so if you multiply the frequency by 2, then the gradients must also end up multiplied by 2.
In your example, the weights are inversely proportional to frequency, so if you have NOT corrected the gradients for frequency already, then just sum them without any weighting at all, because the weight*frequency correction comes out to 1 for all of them.