Question

Suppose w=xy+yz, where x=e2t, y=2+sin(4t), and z=2+cos(7t).

a. use the chain rule to find dwdt as a function of x, y, z, and t. do not rewrite x, y, and z in terms of t, and do not rewrite e2t as x.

Answer 1

basically find the individual derivatives nd put it back into the mother derivative
i.e
dx/dt = 2e^2t
dy/dt= 4cos 4t
dZ/dt = -7sin 7t
now
from
W=xy+yz
dw/dx = y ( by partial differentiation)
dw/dy= x+z
dw/dz= y

now Dw/dt = ⅓{ (dx/dt×dw/dx) + (dy/dt×dw/dy)+(dz/dt×dw/dz)

dw/dt = ⅓{ (2e^2t×y) + (4cos4t×(x+z) + (-7sin7t×y)}
= ⅓{ 2ye^2t + 4(x+z)Cos4t - 7ySin7t}

pheww
that should help...