Finding the Partial Derivatives of a Function of Two Variables

For each function \(z=f(x,y)\), find \(f_{x}(x,y)\) and \(f_{y}(x,y)\).

1. \(f(x,y)=3x^{2}y+2x-3y\)
2. \(f(x,y)=x\;\sin\;y+y\;\sin\;x\)

Solution  (a) To find \(f_{x}(x,y)\), treat \(y\) as a constant in \( f(x,y)=3x^{2}y+2x-3y\) and differentiate with respect to \(x\). The result is \[ f_{x}(x,y)=6\;xy+2 \]

To find \(f_{y}(x,y)\), treat \(x\) as a constant and differentiate with respect to \(y\). The result is \[ f_{y}(x,y)=3x^{2}-3 \]

(b) For \(f(x,y)=x\;\sin\;y+y\;\sin\;x\), we have