find the total differential of the function

Question

f(x,y)= xye^(x+y)

Answer 1

df = ∂f/∂x dx + ∂f/∂y dy

f = xye^(x+y)
df = (ye^(x+y) + xye^(x+y)) dx + (xe^(x+y) + xye^(x+y)) dy
= e^(x+y) [ (x+1)y dx + (y+1)x dy]

or, separating the product a bit,

f = xe^x * ye^y
df = e^x(x+1)*ye^y dx + e^y(y+1)*xe^x dy
= e^x*e^y [(x+1)y dx + (y+1)x dy]