Circle diagram — three links

With three links the method does not change: one simply adds a circle and a new geometric symbol. Example with $e^{\sqrt{x+e^x}}$.

Example — Diagram for $e^{\sqrt{x+e^x}}$

Three links: $x \to x+e^x \to \sqrt{x+e^x} \to e^{\sqrt{x+e^x}}$, with symbols $\square=x$, $\triangle=x+e^x$, $\bigcirc=\sqrt{x+e^x}$.

Three partial derivatives, multiplied and substituted: $D\!\left[e^{\sqrt{x+e^x}}\right] = (1+e^x)\cdot\frac{1}{2\sqrt{x+e^x}}\cdot e^{\sqrt{x+e^x}}.$

Remark — Why it works: Leibniz notation

The circle diagram is the visual version of Leibniz’s chain rule: $\frac{dy}{dx} = \frac{dy}{d\bigcirc}\cdot\frac{d\bigcirc}{d\triangle}\cdot\frac{d\triangle}{d\square}\cdot\frac{d\square}{dx}.$ The geometric symbols in the denominator and numerator “cancel” as in a chain of fractions: this is why the $d$‘s behave as if they were fractions, and why the method produces the correct result.

Links

Topics: Derivatives
Concepts: Composite derivative · Leibniz notation · Chain rule
Skills: Differentiating
People: Gottfried Leibniz