设y=uv,而u=ex,v=sinx, 求dy⼀dx

设: y = uv .......(1)
u = e^x ..........(2)
v = sinx............(3)
求: dy/dx
解法1：(函数乘积求导法)
y ＝uv = e^x sinx........................................................................(4)
dy/dx ＝e^xsinx + e^xcosx = e^x(sinx+cosx)...........(5)
解法2：(全导数求导法)
dy/dx ＝∂y/∂u(du/dx) + ∂y/∂v(dv/dx)
= v(e^x) + u(cosx) = sinx e^x+cosxe^x
= e^x (sinx+cosx) .......................................................(6)

dy/dx=(uv)'=u'v+uv'=(ex)'v+ex(sinx)'=vex+excosx

u=e^x
du/dx =e^x
v=sinx
dv/dx = cosx
------------
y=uv
dy/dx = u.dv/dx + v.du/dx
= (e^x).cosx + sinx . e^x
=(sinx+cosx).e^x