z=(x+4y)ey, x=u, y=ln(v), find ∂z/∂u and ∂z/∂v. The variables are restricted to domains on which the functions are defined.

Step-by-step explanation:

Given the functions z= (x+4y)e^y, x=u, and y =ln(v)

To get ∂z/∂u and ∂z/∂v, we will use the the composite rule formula:

∂z/∂u = ∂z/∂x•dx/du + ∂z/∂y•dy/du

∂z/∂x means we are to differentiate z with respect to x taking y as constant and this is gotten using product.

∂z/∂x = (x+4y)(0)+(1+4y)e^y

∂z/∂x = (1+4y)e^y

dx/du = 1

∂z/∂y = (x+4y)e^y+(x+4)e^y

dy/du = 0

∂z/∂u = (1+4y)e^y • 1 + 0

∂z/∂u = (1+4y)e^y

For ∂z/∂v:

∂z/∂v = ∂z/∂y• dy/dv

∂z/∂y = (x+4y)e^y+(x+4)e^y •(1/v)

∂z/∂y = {xe^y+4ye^y+xe^y+4e^y}•(1/v)

∂z/∂y = 2xe^y/v+4e^y(y+1)/v

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step-by-step explanation: