Chapter 13 - Partial Derivatives - 13.5 The Chain Rule - Exercises Set 13.5 - Page 956: 3

$$\,\,\,\,\frac{{dz}}{{dt}} = \frac{3}{{{t^2}}}\sin \left( {\frac{1}{t}} \right)$$

$$\eqalign{ & z = 3\cos x - \sin xy;\,\,\,\,\,\,\,\,\,x = 1/t,\,\,\,\,\,\,\,y = 3t \cr & \cr & {\text{Calculate the partial derivatives }}\frac{{\partial z}}{{\partial x}}{\text{ and }}\frac{{\partial y}}{{\partial x}} \cr & \,\,\,\,\frac{{\partial z}}{{\partial x}} = \frac{\partial }{{\partial x}}\left( {3\cos x - \sin xy} \right) \cr & \,\,\,\,{\text{treat }}y{\text{ as a constant}} \cr & \,\,\,\,\frac{{\partial z}}{{\partial x}} = 3\frac{\partial }{{\partial x}}\left( {\cos x} \right) - \frac{\partial }{{\partial x}}\left( {\sin xy} \right) \cr & \,\,\,\,\frac{{\partial z}}{{\partial x}} = - 3\sin x - y\cos xy \cr & and \cr & \,\,\,\,\frac{{\partial z}}{{\partial y}} = \frac{\partial }{{\partial y}}\left( {3\cos x - \sin xy} \right) \cr & \,\,\,\,{\text{treat }}x{\text{ as a constant}} \cr & \,\,\,\,\frac{{\partial z}}{{\partial y}} = \frac{\partial }{{\partial y}}\left( {3\cos x} \right) - \frac{\partial }{{\partial y}}\left( {\sin xy} \right) \cr & \,\,\,\,\frac{{\partial z}}{{\partial y}} = - x\cos xy \cr & \cr & {\text{Calculate the ordinary derivatives }}\frac{{dx}}{{dt}}{\text{ and }}\frac{{dy}}{{dt}} \cr & \,\,\,\,\frac{{dx}}{{dt}} = \frac{d}{{dt}}\left( {\frac{1}{t}} \right) = - \frac{1}{{{t^2}}} \cr & \,\,\,\,\frac{{dy}}{{dt}} = \frac{d}{{dt}}\left( {3t} \right) = 3 \cr & \cr & {\text{Use the Theorem 13}}{\text{.5}}{\text{.1}}\,\,\left( {{\text{chain rule for derivatives}}} \right){\text{ see page 950}} \cr & \,\,\,\,\frac{{dz}}{{dt}} = \frac{{\partial z}}{{\partial x}}\frac{{dx}}{{dt}} + \frac{{\partial z}}{{\partial y}}\frac{{dy}}{{dt}} \cr & \,\,\,\,\,{\text{Substitute the derivatives}} \cr & \,\,\,\,\frac{{dz}}{{dt}} = \left( { - 3\sin x - y\cos xy} \right)\left( { - \frac{1}{{{t^2}}}} \right) + \left( { - x\cos xy} \right)\left( 3 \right) \cr & \,\,\,\,{\text{Where }}x = 1/t{\text{ and}}\,\,\,\,\,y = 3t \cr & \,\,\,\,\frac{{dz}}{{dt}} = \left( { - 3\sin \left( {\frac{1}{t}} \right) - 3t\cos \left[ {\left( {\frac{1}{t}} \right)\left( {3t} \right)} \right]} \right)\left( { - \frac{1}{{{t^2}}}} \right) + \left( { - \frac{1}{t}\cos \left[ {\left( {\frac{1}{t}} \right)\left( {3t} \right)} \right]} \right)\left( 3 \right) \cr & \,\,\,\,\frac{{dz}}{{dt}} = \frac{3}{{{t^2}}}\sin \left( {\frac{1}{t}} \right) + \frac{3}{t}\cos 3 + - \frac{3}{t}\cos 3 \cr & \,\,\,\,\frac{{dz}}{{dt}} = \frac{3}{{{t^2}}}\sin \left( {\frac{1}{t}} \right) \cr} $$