Derivative of e to the cos(2x)

Statement

Using the circle diagram, compute $D[e^{\cos(2x)}]$.

Solution

Three rings: $x \to 2x \to \cos(2x) \to e^{\cos(2x)}$. Partial derivatives: $2$ (for $2\square$), $-\sin\triangle$ (for $\cos\triangle$), $e^{\bigcirc}$ (for $e^{\bigcirc}$). Multiplying and substituting: $D[e^{\cos(2x)}] = e^{\cos(2x)}\cdot(-\sin(2x))\cdot 2 = \boxed{-2\sin(2x)\,e^{\cos(2x)}}.$

Links

Topics: Derivatives
Concepts: Composite derivative · Chain rule
Skills: Differentiating
Exercise type: Derivative computation