Derivative of arctan(√x)

Statement

Compute $D[\arctan(\sqrt{x})]$ with the circle diagram.

Solution

Two rings: $x \to \sqrt{x} \to \arctan(\sqrt{x})$. Partial derivatives: $\frac{1}{2\sqrt{\square}}$ (for $\sqrt{\square}$), $\frac{1}{1+\triangle^2}$ (for $\arctan\triangle$). Multiplying and substituting ($\triangle = \sqrt{x}$, so $\triangle^2 = x$): $D[\arctan(\sqrt{x})] = \frac{1}{1+x}\cdot\frac{1}{2\sqrt{x}} = \boxed{\frac{1}{2\sqrt{x}\,(1+x)}}.$

Links

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