Answer
$$\frac{{dw}}{{dt}} = - \sin \left( {3{t^4}} \right)\sin \left( t \right) + 12{t^3}\cos \left( t \right)\cos \left( {3{t^4}} \right)$$
Work Step by Step
$$\eqalign{
& {\text{we have }}w = \cos 2x\sin 3y,{\text{ with }}x = t/2{\text{ and }}y = {t^4} \cr
& {\text{find }}\frac{{dw}}{{dt}}{\text{ using the theorem 12}}{\text{.7 }}\left( {{\text{see page 908}}} \right) \cr
& \frac{{dw}}{{dt}} = \frac{{\partial w}}{{\partial x}}\frac{{dx}}{{dt}} + \frac{{\partial w}}{{\partial y}}\frac{{dy}}{{dt}} \cr
& {\text{find the partial derivatives }}\frac{{\partial w}}{{\partial x}}{\text{ and }}\frac{{\partial w}}{{\partial y}} \cr
& \frac{{\partial w}}{{\partial x}} = \frac{\partial }{{\partial x}}\left[ {\cos 2x\sin 3y} \right] \cr
& \frac{{\partial w}}{{\partial x}} = \sin 3y\frac{\partial }{{\partial x}}\left[ {\cos 2x} \right] \cr
& \frac{{\partial w}}{{\partial x}} = - 2\sin 3y\sin 2x \cr
& and \cr
& \frac{{\partial w}}{{\partial y}} = \frac{\partial }{{\partial y}}\left[ {\cos 2x\sin 3y} \right] \cr
& \frac{{\partial w}}{{\partial y}} = \cos 2x\frac{\partial }{{\partial y}}\left[ {\sin 3y} \right] \cr
& \frac{{\partial w}}{{\partial y}} = 3\cos 2x\cos 3y \cr
& {\text{find the partial derivatives }}\frac{{dx}}{{dt}}{\text{ and }}\frac{{dy}}{{dt}} \cr
& \frac{{dx}}{{dt}} = \frac{d}{{dt}}\left[ {\frac{t}{2}} \right] = \frac{1}{2} \cr
& \frac{{dy}}{{dt}} = \frac{d}{{dt}}\left[ {{t^4}} \right] = 4{t^3} \cr
& \cr
& {\text{substitute the results of the derivatives in the equation of the theorem 12}}{\text{.7}} \cr
& \frac{{dw}}{{dt}} = \frac{{\partial w}}{{\partial x}}\frac{{dx}}{{dt}} + \frac{{\partial w}}{{\partial y}}\frac{{dy}}{{dt}} \cr
& \frac{{dw}}{{dt}} = \left( { - 2\sin 3y\sin 2x} \right)\left( {\frac{1}{2}} \right) + \left( {3\cos 2x\cos 3y} \right)\left( {4{t^3}} \right) \cr
& \frac{{dw}}{{dt}} = - \sin 3y\sin 2x + 12{t^3}\cos 2x\cos 3y \cr
& {\text{replace }}x = t/2{\text{ and }}y = {t^4} \cr
& \frac{{dw}}{{dt}} = - \sin \left( {3{t^4}} \right)\sin \left( {2\left( {\frac{t}{2}} \right)} \right) + 12{t^3}\cos \left( {2\left( {\frac{t}{2}} \right)} \right)\cos \left( {3{t^4}} \right) \cr
& {\text{simplifying}} \cr
& \frac{{dw}}{{dt}} = - \sin \left( {3{t^4}} \right)\sin \left( t \right) + 12{t^3}\cos \left( t \right)\cos \left( {3{t^4}} \right) \cr} $$
