Answer
a. $x$ first
b. $y$ first
c. $x$ first
d. $x$ first
e. $y$ first
f. $y$ first
Work Step by Step
a. $f(x,y)=x\sin y+e^y$
We differentiate with respect to $x$ first
as there is only one term having $x$ while for $y$ there is exponential which never ends
b. $f(x, y) = \frac{1}{x}$
We differentiate with respect to $y$ first
because there is no $y$ variable.
c. $f(x, y) = y + (\frac{x}{y})$
We differentiate with respect to $x$ first
as there is only one term with the variable $x$.
d. $f(x, y) = y + x^{2}y + 4y^{3} - \ln ( y^{2} + 1)$
We differentiate with respect to $x$ first as there is only one term having the variable $x$.
e. $f(x, y) = x^{2} + 5xy + \sin x + 7e^{x}$
We differentiate with respect to $y$ first as there is only one term with the variable $y$.
Why we choose the variable with minimum terms? Because the rest would be constant so derivation of constant is zero, it would reduce the term for next derivation . Remember this is only for continuous functions.
f. $f(x, y) = x \ln xy$
We differentiate with respect to $y$ first as there is only one term containing $y$.
