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The model tracks a small input change $dx$ through successive local linear approximations. First, the inner function $h(x)=x^2$ scales the input change by its derivative $h'(x)=2x$.

Conditions: Functions $h$ and $g$ are differentiable at the relevant points.; The input change $dx$ is infinitesimal.; The outer derivative is evaluated at the inner function's output $h(x)$, not at $x$.

The model tracks a small input change $dx$ through successive local linear approximations. First, the inner function $h(x)=x^2$ scales the input change by its derivative $h'(x)=2x$.

Conditions: Functions $h$ and $g$ are differentiable at the relevant points.; The input change $dx$ is infinitesimal.; The outer derivative is evaluated at the inner function's output $h(x)$, not at $x$.

Leibniz notation $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$ resembles algebraic cancellation, but derivatives are limits, not ordinary fractions. The rigorous justification relies on expressing increments exactly as $\Delta y = [g'(u)+\epsilon]\Delta u$ with $\epsilon \to 0$.

Conditions: Functions are differentiable.; The inner derivative $du/dx$ can be zero.; The argument requires rigorous limit definitions, not informal algebra.

Leibniz notation $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$ resembles algebraic cancellation, but derivatives are limits, not ordinary fractions. The rigorous justification relies on expressing increments exactly as $\Delta y = [g'(u)+\epsilon]\Delta u$ with $\epsilon \to 0$.

Conditions: Functions are differentiable.; The inner derivative $du/dx$ can be zero.; The argument requires rigorous limit definitions, not informal algebra.