| Page 711 #7 | Evaluate \(f_x (x, y)\) and \(f_y (x, y)\) at the indicated point \((\frac{\pi}{3},\frac{\pi}{3})\).

\[f(x,y) = sin(y)cos(x)\]

Partial Derivative (Page 703)

For \(f_x\) treat \(y\) as a constant

Take constant out \((a*f') = a*f'\)

\[f_x = sin(y)(-sin(x))\]

For \(f_y\) treat \(x\) as a constant

Take constant out \((a*f') = a*f'\) \[f_y= cos(y)cos(x)\]

Trigonometry Review

sin(x) cos(x) tan(x)
0 = 0 0 =1 0=0
\(\frac{\pi}{6} = \frac{1}{2}\) \(\frac{\pi}{6}= \frac{\sqrt3}{2}\) \(\frac{\pi}{6}=\frac{1}{\sqrt3}\)
\(\frac{\pi}{4} = \frac{\sqrt2}{2}\) \(\frac{\pi}{4}=\frac{\sqrt2}{2}\) \(\frac{\pi}{4}=1\)
\(\frac{\pi}{3}= \frac{\sqrt3}{2}\) \(\frac{\pi}{3}= \frac{1}{2}\) \(\frac{\pi}{3}=\sqrt3\)
\(\frac{\pi}{2}=1\) \(\frac{\pi}{2}= 0\) \(\frac{\pi}{2}\)=undefined

Using the values above, we can substitute the indicated point \((\frac{\pi}{3},\frac{\pi}{3})\) into \(f_x\) and \(f_y\)

\(f_x\)

\[f_x = sin(y)(-sin(x))\] \[f_x = sin(\frac{\pi}{3})(-sin(\frac{\pi}{3}))\] \[f_x = \frac{\sqrt3}{2} (-\frac{\sqrt3}{2})\] \[f_x=\frac{-3}{4}\]

\(f_y\)

\[f_y= cos(y)cos(x)\]

\[f_y= cos(\frac{\pi}{3})cos(\frac{\pi}{3})\] \[f_y= \frac{1}{2}*\frac{1}{2}\] \[f_y=\frac{1}{4}\]