The Chain Rule

Differentiate $y = \sin^3(\ln(\cos x))$ with respect to $x$.

A particle moves along a line so that its position at time $t$ (in seconds) is given by $s(t) = \cos\left(\sqrt{t^2 + 1}\right)$ meters. Find the velocity function $v(t) = s'(t)$.

A spherical balloon is being inflated. Its radius at time $t$ seconds is $r(t)=\sqrt{2t+1}$ cm. The volume of a sphere is $V=\frac{4}{3}\pi r^3$. Determine the rate of change of volume with respect to time, $\frac{dV}{dt}$, as an expression in $t$.