Circle diagram — arctan(1/x)

A second example with two links, in which the first link is the function $1/x$: it shows how the method also handles derivatives with a sign and with denominators.

Example — Diagram for $\arctan(1/x)$

Composition $x \to \frac{1}{x} \to \arctan\frac{1}{x}$: the derivative of $\frac{1}{\square}$ is $-\frac{1}{\square^2}$, the derivative of $\arctan\triangle$ is $\frac{1}{1+\triangle^2}$.

Multiplication and substitution ($\square = x$, $\triangle = 1/x$): $D\!\left[\arctan\!\tfrac{1}{x}\right] = \left(-\frac{1}{\square^2}\right)\cdot\frac{1}{1+\triangle^2} = -\frac{1}{x^2}\cdot\frac{1}{1+(1/x)^2} = -\frac{1}{x^2+1}.$

Links

Topics: Derivatives
Concepts: Composite derivative · Chain rule
Functions: Arctangent · Inverse trigonometric functions
Skills: Differentiating