Assuming that the equation
e^z − y + xz = 0
defines z as a function of x and y with z(1, 1) = 0 and
that this function has a Taylor series about (1, 1), find
the terms up to and including order 2 in this Taylor
series and use this to find an approximate value for z
when x = 1.01 and y = 0.9.
Any help is much appreciated. Thanks
These two articles should point the way for you
math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Multivariable_Calculus/3%3A_Topics_in_Partial_Derivatives/Taylor__Polynomials_of_Functions_of_Two_Variables
www.math.cmu.edu/~cnewstea/teaching/old/teaching/21-256/implicit-partial-diff.pdf