Partial derivatives of a multivariate function

The problem, from first principles

A single-variable derivative answers "how does output change as this one input changes." But a real loss function depends on hundreds, thousands, or millions of numbers at once, every weight and bias in a network. You can't ask "how does the loss change" without saying which number you're wiggling, so the natural move is to ask the single-variable question over and over, once per number, holding everything else perfectly still each time.

That's a partial derivative. And collecting every one of those answers into a single vector, one entry per parameter, gives you the gradient, the object every optimizer in this curriculum (04-gd-step onward) actually steps against.

From theory to code

Theory reduces a partial derivative to 01-derivatives-first-principles's central difference, applied to just one coordinate of a multi-dimensional input while every other coordinate stays fixed. Implement that first, then collect it across every coordinate to build the full gradient.

Implement partial_derivative(f, x, index, eps=1e-5) and gradient(f, x, eps=1e-5) against that reasoning. The signatures and docstrings are already in the editor.

Constraints

• f takes a NumPy array x and returns a scalar.
• partial_derivative must not mutate the caller's x, copy before nudging.
• gradient's output has the same shape as x, one partial derivative per coordinate, in the same order.

Hints

Open one at a time. Each gives away a little more than the last.