Chapter 13 - Partial Derivatives - 13.5 The Chain Rule - Exercises Set 13.5 - Page 957: 23

$$\eqalign{ & \frac{{\partial T}}{{\partial r}} = 3{r^2}{\cos ^2}\theta \sin \theta - 4{r^3}\cos \theta {\sin ^3}\theta \cr & \frac{{\partial T}}{{\partial r}} = - 2{r^2}\cos \theta \sin \theta + {r^3}{\sin ^4}\theta + {r^3}{\cos ^3}\theta - 3{r^3}{\cos ^2}\theta {\sin ^2}\theta \cr} $$

$$\eqalign{ & T = {x^2}y - x{y^3} + 2;\,\,\,\,\,\,\,\,x = r\cos \theta ,\,\,\,\,\,\,\,y = r\sin \theta \cr & \cr & {\text{Calculate the partial derivatives }}\frac{{\partial T}}{{\partial x}}{\text{ and }}\frac{{\partial T}}{{\partial y}} \cr & \frac{{\partial T}}{{\partial x}} = \frac{\partial }{{\partial x}}\left[ {{x^2}y - x{y^3} + 2} \right] \cr & {\text{Treat }}y{\text{ as a constant}} \cr & \frac{{\partial T}}{{\partial x}} = 2xy - {y^3} \cr & \cr & \frac{{\partial T}}{{\partial y}} = \frac{\partial }{{\partial y}}\left[ {{x^2}y - x{y^3} + 2} \right] \cr & {\text{Treat }}x{\text{ as a constant}} \cr & \frac{{\partial T}}{{\partial y}} = {x^2} - 3x{y^2} \cr & \cr & {\text{Calculate the partial derivatives }}\frac{{\partial x}}{{\partial r}}{\text{,}}\,\,\frac{{\partial x}}{{\partial \theta }},\,\,\,\,\frac{{\partial y}}{{\partial r}}{\text{ and }}\frac{{\partial y}}{{\partial \theta }} \cr & \frac{{\partial x}}{{\partial r}} = \frac{\partial }{{\partial r}}\left[ {r\cos \theta } \right] = \cos \theta \cr & \frac{{\partial x}}{{\partial \theta }} = \frac{\partial }{{\partial \theta }}\left[ {r\cos \theta } \right] = - r\sin \theta \cr & \frac{{\partial y}}{{\partial r}} = \frac{\partial }{{\partial r}}\left[ {r\sin \theta } \right] = \sin \theta \cr & \frac{{\partial y}}{{\partial \theta }} = \frac{\partial }{{\partial \theta }}\left[ {r\sin \theta } \right] = r\cos \theta \cr & \cr & {\text{Use the theorem 13}}{\text{.5}}{\text{.2 }}\left( {{\text{see page 952}}} \right){\text{ to find }}\frac{{\partial T}}{{\partial r}}{\text{ and }}\frac{{\partial T}}{{\partial \theta }} \cr & \frac{{\partial T}}{{\partial r}} = \frac{{\partial T}}{{\partial x}}\frac{{\partial x}}{{\partial r}} + \frac{{\partial T}}{{\partial y}}\frac{{\partial y}}{{\partial r}} \cr & {\text{substitute the derivatives}} \cr & \frac{{\partial T}}{{\partial r}} = \left( {2xy - {y^3}} \right)\left( {\cos \theta } \right) + \left( {{x^2} - 3x{y^2}} \right)\left( {\sin \theta } \right) \cr & {\text{where }}x = r\cos \theta ,\,\,\,\,\,\,\,y = r\sin \theta \cr & \frac{{\partial T}}{{\partial r}} = \left( {2\left( {r\cos \theta } \right)\left( {r\sin \theta } \right) - {{\left( {r\sin \theta } \right)}^3}} \right)\left( {\cos \theta } \right) \cr & \,\,\,\,\,\,\,\,\,\,\, + \left( {{{\left( {r\cos \theta } \right)}^2} - 3\left( {r\cos \theta } \right){{\left( {r\sin \theta } \right)}^2}} \right)\left( {\sin \theta } \right) \cr & {\text{simplifying}} \cr & \frac{{\partial T}}{{\partial r}} = \left( {2{r^2}\cos \theta \sin \theta - {r^3}{{\sin }^3}\theta } \right)\left( {\cos \theta } \right) + \left( {{r^2}{{\cos }^2}\theta - 3{r^3}\cos \theta {{\sin }^2}\theta } \right)\left( {\sin \theta } \right) \cr & \frac{{\partial T}}{{\partial r}} = 2{r^2}{\cos ^2}\theta \sin \theta - {r^3}\cos \theta {\sin ^3}\theta + {r^2}\sin \theta {\cos ^2}\theta - 3{r^3}\cos \theta {\sin ^3}\theta \cr & \frac{{\partial T}}{{\partial r}} = 3{r^2}{\cos ^2}\theta \sin \theta - 4{r^3}\cos \theta {\sin ^3}\theta \cr & \cr & and \cr & \cr & \frac{{\partial T}}{{\partial \theta }} = \frac{{\partial T}}{{\partial x}}\frac{{\partial x}}{{\partial \theta }} + \frac{{\partial T}}{{\partial y}}\frac{{\partial y}}{{\partial \theta }} \cr & {\text{substitute the derivatives}} \cr & \frac{{\partial T}}{{\partial r}} = \left( {2xy - {y^3}} \right)\left( { - r\sin \theta } \right) + \left( {{x^2} - 3x{y^2}} \right)\left( {r\cos \theta } \right) \cr & {\text{where }}x = r\cos \theta ,\,\,\,\,\,\,\,y = r\sin \theta \cr & \frac{{\partial T}}{{\partial r}} = \left( {2\left( {r\cos \theta } \right)\left( {r\sin \theta } \right) - {{\left( {r\sin \theta } \right)}^3}} \right)\left( { - r\sin \theta } \right) \cr & \,\,\,\,\,\,\,\,\,\,\, + \left( {{{\left( {r\cos \theta } \right)}^2} - 3\left( {r\cos \theta } \right){{\left( {r\sin \theta } \right)}^2}} \right)\left( {r\cos \theta } \right) \cr & {\text{simplifying}} \cr & \frac{{\partial T}}{{\partial r}} = - 2{r^2}\cos \theta \sin \theta + {r^3}{\sin ^4}\theta + {r^3}{\cos ^3}\theta - 3{r^3}{\cos ^2}\theta {\sin ^2}\theta \cr} $$