The chain rule as a pipeline

The derivative $f'(x)$ is the slope of $f$ at $x$: how much the output changes per tiny change in input. Rules worth knowing cold: power rule $\frac{d}{dx}x^n = nx^{n-1}$, exponentials $\frac{d}{dx}e^x = e^x$, and the sigmoid's tidy derivative $\sigma'(x) = \sigma(x)(1-\sigma(x))$.

The chain rule: if $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x))\cdot g'(x)$. Wiggle $x$; it moves $g$ by $g'$; that moves $f$ by $f'$ times as much. Rates multiply along the chain.

A neural network is a long chain of functions. Backpropagation is nothing more than applying the chain rule from the loss backwards through every step, reusing intermediate results.