Chain Rule¶

Formula¶

\[ \frac{d}{dx} f(g(x)) = f'(g(x))\,g'(x) \]

Parameters¶

\(g(x)\): inner function
\(f(\cdot)\): outer function

What it means¶

The derivative of a composition is the product of the outer derivative (evaluated at the inner function) and the inner derivative.

What it's used for¶

Differentiating nested expressions.
Backpropagation in neural networks.

Key properties¶

Applies repeatedly through deep compositions.
Generalizes to Jacobians in multivariable settings.

Common gotchas¶

Forgetting to evaluate the outer derivative at \(g(x)\).
Missing inner derivatives in long compositions.

Example¶

If \(f(u)=u^2\) and \(g(x)=\sin x\), then \(\frac{d}{dx}(\sin x)^2 = 2\sin x \cos x\).

How to Compute (Pseudocode)¶

Input: outer function f, inner function g, point x
Output: derivative of h(x) = f(g(x)) at x

u <- g(x)
inner_deriv <- g'(x)
outer_deriv <- f'(u)
return outer_deriv * inner_deriv

Complexity¶

Time: \(O(1)\) high-level composition steps, plus the costs of evaluating \(g(x)\), \(g'(x)\), and \(f'(g(x))\)
Space: \(O(1)\)
Assumptions: This is the scalar single-composition case; nested compositions scale with the number of layers/operations