Answer
$f_{xx}=-\frac{a^{2}}{({ax+by})^{2}}$
$f_{yy}=-\frac{b^{2}}{({ax+by})^{2}}$
$f_{xy}=f_{yx}=-\frac{ab}{({ax+by})^{2}}$
Work Step by Step
Use the derivative of natural log to find first partial derivatives. For x, treat the variable y as a constant, and vice versa.
$f_{x}=\frac{1}{ax+by}\times(a) = \frac{a}{ax+by}$
$f_{y}=\frac{1}{ax+by}\times(b) = \frac{b}{ax+by}$
Then take the derivative of the first order partial derivatives to find second partial derivatives.
$f_{xx}=-\frac{a^{2}}{({ax+by})^{2}}$
$f_{yy}=-\frac{b^{2}}{({ax+by})^{2}}$
Second partial derivatives of first order partial derivative of x with respect to y and y with respect to x are the same:
$f_{xy}=f_{yx}=-\frac{ab}{({ax+by})^{2}}$
