How does the three-tiered number line model demonstrate the chain rule for the composite function $g(h(x)) = \sin(x^2)$?

The chain rule holds whenever both the inner function $h$ is differentiable at $x$ and the outer function $g$ is differentiable at $h(x)$. If $h'(x)=0$, the formula $g'(h(x))h'(x)$ correctly yields 0.

Conditions: $h$ is differentiable at $x$; $g$ is differentiable at $h(x)$; No requirement for $h'(x) \neq 0$

By tracking a small input change $dx$ through nested mappings $x \to h(x) \to g(h(x))$. For $x=1.5$, $h=x^2$, and $g=\sin$, the inner rate is $h'(1.5)=3$, scaling $dx$ to $dh \approx 3dx$.

Conditions: Inner function $h(x)$ is differentiable at $x$; Outer function $g(u)$ is differentiable at $u=h(x)$; Example uses $h(x)=x^2$ and $g(u)=\sin u$ near $x=1.5$

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.

The outer derivative $g'$ must be evaluated at the output of the inner function, i.e., at $h(x)$, not at the original input $x$. This reflects the sequential nature of composition: the change in $x$ affects $h$, and the sensitivity of $g$ depends on the current value of $h$.

Conditions: Computing derivative of $g(h(x))$; Applying chain rule $g'(h(x))h'(x)$