### Automatic differentiation in  C++ ###

Day 8: Forward mode implementation part 1

I've figured out how to implement the forward mode for the simple cases: 
Case 1: input x is a vector and output y is a real number.
The second case is just a generalization of the output (y is also a vector).

I had some cool small examples of the manual execution and values of the gradients trace, but it got deleted somehow 🥲 I'll do it again and share it the next day!

On a side note, I'm still trying to get used to vim motions, my mouse reflections still kicks in whenever I want to change files or scroll through the code 😂 I'm liking the hjkl movements as well as the basic insertion and cursor movements shortcuts, vey nice!

When treating the input as a single input vector, forward mode would compute dy/dx in one pass, and if you treat it as multiple inputs, the forward mode will compute it dy/dx_i for each value in that input, thus requiring multiple forward passing. Implementation wise, I'll have to think which way is better.

Quick update, I've decided to make the implementation consider the input vector as multiple inputs (x_1, x_2..., x_n), seems to be what's FM is intended to be implemented as. I'll finish it today hoepfully and share the push the code to git soon!

In forward mode, it seems that each forward pass can compute the gradient of the output with respect to only one input at a time. So if the input is a vector, would that be considered as 1 input or would each value in that vector be considered as 1 input (so x has n inputs basically)?