Derivative of ln(sin(x³))

Statement

Using the circle diagram, compute $D[\ln(\sin(x^3))]$.

Solution

Three rings: $x \to x^3 \to \sin(x^3) \to \ln(\sin(x^3))$. The partial derivatives are $3x^2$ (for $\square^3$), $\cos\triangle$ (for $\sin\triangle$), $\frac{1}{\bigcirc}$ (for $\ln\bigcirc$). Multiplying and substituting: $D[\ln(\sin(x^3))] = \frac{1}{\sin(x^3)}\cdot\cos(x^3)\cdot 3x^2 = \boxed{\frac{3x^2\cos(x^3)}{\sin(x^3)}}.$

Links

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