The chain rule and logarithms practice question 17

Count the layers in $\sin^{3}\!\left(\pi x^{2}\right)$. Which operation is done last, and which sits innermost?

Three layers: a cube on the outside, then sine, then $\pi x^{2}$ inside. Differentiate from the outside in, one factor per layer.

$\dfrac{d}{dx}\sin^{3}u = 3\sin^{2}u\,\cos u\cdot u'$, with $u = \pi x^{2}$. The last factor, $u'$, is the one most often lost.