Question

find the total differential dz for \[z= x^y\] find the total differential dz for \[z= x^y\] @Mathematics

Answer 1

\[dz={∂z \over ∂x} dx + {∂z \over ∂y} dy \]\[ {∂z \over ∂x} = yx^{y-1} = {y \over x }x^y = {y \over x} z\]\[ {∂z \over ∂y} = {∂(x^y) \over ∂y}= ?\]

Answer 2

\[\frac{\partial z}{\partial y}=\frac{\partial (x^y)}{\partial y}=x^y\ln{x}\]

Answer 3

is there an intermediate step? i cant see how you know/can prove that