Finding the Partial Derivatives of a Function at a Point \(( x_{0},y_{0})\)

For \(f(x,y)=x\;\sin\;y+y\;\sin\;x\), find \(f_{x}\left(\dfrac{\pi }{3},\dfrac{\pi }{ 6}\right)\) and \(f_{y}\left( \dfrac{\pi }{3},\dfrac{\pi }{6}\right)\).

Solution  We use the results from Example 1(b): \[ \begin{eqnarray*} f_{x}(x,y) &=&\sin\;y+y\;\cos\;x\\[3pt] f_{x}\left( \dfrac{\pi }{3},\dfrac{\pi }{6}\right) &=&\sin \dfrac{\pi }{6}+\dfrac{\pi }{6}\cos \dfrac{\pi }{3}=\dfrac{1}{2}+\dfrac{\pi }{6}\left( \dfrac{1}{2}\right) =\dfrac{6+\pi }{12}\\ f_{y}(x,y)&=&x\;\cos\;y+\;\sin\;x\\[3pt] f_{y}\left( \dfrac{\pi }{3},\dfrac{\pi }{6}\right) &=&\dfrac{\pi }{3}\cos \dfrac{\pi }{6}+\sin \dfrac{\pi }{3}=\dfrac{\pi }{3}\left( \dfrac{\sqrt{3}}{2}\right) +\dfrac{\sqrt{3}}{2 }=\dfrac{( \pi +3) \sqrt{3}}{6} \end{eqnarray*} \]