Visualizing the chain rule and product rule | Chapter 4, Essence of calculus

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 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 $x=1.5$

The constant multiple rule is a special case of the product rule where one factor is a constant $c$. Geometrically, multiplying a function by a constant scales its output heights (and thus its area strips) by $c$.

Conditions: $c$ is a constant.; The function $f$ is differentiable.

The video advises identifying the structural hierarchy of the expression first. Determine if the outermost operation is a sum, a product, or a composition.

Conditions: The expression involves multiple operations (sums, products, compositions).; The learner aims to apply differentiation rules correctly.

The term vanishes because differentiability ensures that both $\Delta f$ and $\Delta g$ are proportional to the input increment $h$ (i.e., of order $O(h)$). When divided by $h$, the product $\frac{\Delta f \Delta g}{h}$ becomes an expression of order $O(h^2)/h = O(h)$, which approaches zero as $h \to 0$.

Conditions: Functions $f$ and $g$ are differentiable at the point of interest; Input increment $h$ approaches zero

The corner term represents the product of two small changes, $\Delta f \Delta g$. Because the functions are differentiable, both $\Delta f$ and $\Delta g$ are of order $h$ (proportional to the input increment).

Conditions: Functions $f$ and $g$ are differentiable at the point.; The input increment $h$ approaches zero.; The geometric model assumes a rectangle with sides $f(x)$ and $g(x)$.

By visualizing the functions $f$ and $g$ as vertical heights stacked on top of each other, a common input increment $dx$ produces separate changes $\Delta f$ and $\Delta g$. The total change in the sum $f+g$ is exactly the sum of these individual changes.

Conditions: Functions $f$ and $g$ are differentiable.; The input increment is common to both functions.

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 sum rule is visualized by stacking vertical heights, where an input increment causes additive changes in height ($(f+g)'=f'+g'$). In contrast, the product rule is visualized as an expanding rectangle with sides $f(x)$ and $g(x)$, where the area change consists of two rectangular strips and a small corner square ($\Delta(fg) = g\Delta f + f\Delta g + \Delta f \Delta g$).

Conditions: Comparing linear combination vs multiplicative interaction; Using geometric models for differentiation

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 $g'(h(x))h'(x)$