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I am learning about Apache MXNet and I have encountered the following simple example:

https://mxnet.apache.org/versions/1.9.1/api/python/docs/tutorials/getting-started/crash-course/3-autograd.html

Specifically, this tutorial contains the following code snippet:

from mxnet import nd
from mxnet import autograd
    
x = nd.array([[1, 2], [3, 4]])
x.attach_grad()
with autograd.record():
    y = 2* x * x

y.backward()
print(x.grad)

The tutorial explains,

The derivative of 2x^2 with respect to x is 4x, thus x.grad = [[4, 8], [12, 16]].

This would make sense to me if x were a real variable, but x is not a real variable, it's a 2x2 matrix. I thought that because the function y=y(x) takes a 2x2 matrix and returns a 2x2 matrix, the rules of single variable calculus do not apply.

Mathematically, why is x.grad equal to 4*x?

firstname gklsodascb
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1 Answers1

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It's off-topic but I'd still like to answer.

The derivation of 2x² is 4x. You can calculate that by multiplying the exponent with the number in front of the x and reducing the exponent by 1.

Now you can multiply every value (x) in the matrizes with 4. Which results in the numbers: 4, 8 - 12, 16

All values, no matter if it's one or two dimensional are x in that case. You apply this function to every single value there.

SickerDude43
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