Question

Suppose z=x^2*sin(y), x=2s^2−4t^2, y=4st.
A. Use the chain rule to find ∂z/∂s and ∂z/∂t as functions of x, y, s and t.

Answer 1

@experimentX

Answer 2

@robtobey

Answer 3

@calculusfunctions

Answer 4

dz/ds = dz/dx*dx/ds + dz/dy*dy/ds

Answer 5

dz/ds = 2x*sin(y)*(4s) + x^2*cos(y)*4t

Answer 6

dz/dt = dz/dx*dx/dt + dz/dy*dy/dt

Answer 7

dz/dt = -2x*sin(y)*8t + x^2*cos(y)*4s

Answer 8

B. Find the numerical values of ∂z∂s and ∂z∂t when (s,t)=(−5,−1).
∂z∂s(−5,−1)=
∂z∂t(−5,−1)=

Answer 9

substitute 2s^2 - 4t^2 for x,
substitute 4st for y

Answer 10

dz/ds in terms of s and t = 2*(2s^2 - 4t^2)*sin(4st)*(4s) + (2s^2 - 4t^2)^2*cos(4st)*(4s)

Answer 11

Plug in -5 and -1 for t and s Respectively

Answer 12

dz/ds (-5,-1) = -1840sin(20) - 8464(cos(20))

Answer 13

dz/dt (-5, -1) = 8*92*sin(20) - 20*(46^2)*cos(20)

Answer 14

dz/dt (-5, -1) = 736sin(20) - 42320cos(20)