y=-e^{sin(pi/x)}sin(pi x)
dy/dx=d/dx[-e^{sin(pi/x)}sin(pi x)]
dy/dx=d/dx[-e^{sin(pi/x)}]sin(pi x) + d/dx[ sin(pi x)](- e^{sin(pi/x)})
d/dx[ sin(pi x)](- e^{sin(pi/x)})= pi cos(pix)(- e^{sin(pi/x)})
d/dx[ sin(pi x)](- e^{sin(pi/x)})=- pie^{sin(pi/x)} cos(pix)
Chain rule time: alpha = sin(pi/x), beta = 1/x
d/dx[-e^{sin(pi/x)}] = d/{dalpha}[-e^{alpha}]d/{dbeta}[sin(pi beta)]d/dx[1/x]
d/dx[-e^{sin(pi/x)}] = -e^{alpha} times pi cos(pi beta) times (-1/x^2)
d/dx[-e^{sin(pi/x)}] = -e^{sin(pi/x)} times pi cos(pi/x) times (-1/x^2)
d/dx[-e^{sin(pi/x)}] = {pi e^{sin(pi/x)} cos(pi/x)}/{x^2}
--
d/dx[-e^{sin(pi/x)}]sin(pi x) = {pi e^{sin(pi/x)} sin(pi x)cos(pi/x)}/{x^2}
Adding the two together we get:
dy/dx={pi e^{sin(pi/x)} sin(pi x)cos(pi/x)}/{x^2}- pie^{sin(pi/x)} cos(pix)