91影视

Already we have $f(x,y)=\cos 鈦�\left(xy\right)$at position $P=\left(x_0,y_0\right)=(蟺,鈭�3)$

The line of tangent equation is

$f_x\left(x_0,y_0\right)\left(x鈭�x_0\right)+f_y\left(x_0,y_0\right)\left(y鈭�y_0\right)=z鈭�f\left(x_0,y_0\right)$

Equation $1$

$f_x(蟺,鈭�3)(x鈭�蟺)+f_y(蟺,鈭�3)(y+3)=z鈭�f(蟺,鈭�3)$

Then,

$f_x\left(x_0,y_0\right)={\left.\frac{d}{dx}f(x,y)\right|}_{\left(x_0,y_0\right)}={\left.\frac{d}{dx}\cos 鈦�\left(xy\right)\right|}_{\left(x_0,y_0\right)}$

$f_x\left(x_0,y_0\right)={\left.\frac{\mathrm{鈭�}}{\mathrm{鈭�}x}(\cos 鈦�(xy\left)\right)\frac{\mathrm{鈭�}}{\mathrm{鈭�}x}\left(xy\right)\right|}_{\left(x_0,y_0\right)}$

$\begin{array}{r}{f}_x\left(x_0,y_0\right)=鈭�{\left.\sin 鈦�\left(xy\right)脳y\right|}_{\left(x_0,y_0\right)}\\ f_x\left(x_0,y_0\right)=鈭�y脳\sin 鈦�(xy{)}_{\left(x_0,y_0\right)}\end{array}$

$f_x(蟺,鈭�3)=鈭�{\left.y脳\sin 鈦�\left(xy\right)\right|}_{(蟺,鈭�3)}$

$f_x(蟺,鈭�3)=鈭�(鈭�3)脳\sin 鈦�(蟺脳(鈭�3\left)\right)$

$f_x(蟺,鈭�3)=3脳鈭�\sin 鈦�\left(3蟺\right)$

$f_x(蟺,鈭�3)=3脳0$

$(鈭�\sin 鈦�(3蟺)=0)$

Equation$2$

$f_x(蟺,鈭�3)=0$

$f_y\left(x_0,y_0\right)={\left.\frac{d}{dy}f(x,y)\right|}_{\left(x_0,3_0\right)}={\left.\frac{d}{dy}\cos 鈦�\left(xy\right)\right|}_{\left(x_0,y_0\right)}$

$f_y\left(x_0,y_0\right)={\left.\frac{\mathrm{鈭�}}{\mathrm{鈭�}y}(\cos 鈦�(xy\left)\right)\frac{\mathrm{鈭�}}{\mathrm{鈭�}y}\left(xy\right)\right|}_{\left(x_0,y_0\right)}$

$\begin{array}{r}{f}_y\left(x_0,y_0\right)=鈭�{\left.\sin 鈦�\left(xy\right)脳x\right|}_{\left(x_0,x_0\right)}\\ f_y\left(x_0,y_0\right)=鈭�{\left.x脳\sin 鈦�\left(xy\right)\right|}_{\left(x_0,y_5\right)}\end{array}$

$f_y(蟺,鈭�3)=鈭�{\left.x脳\sin 鈦�\left(xy\right)\right|}_{(蟺,鈭�3)}$

$f_y(蟺,鈭�3)=鈭�\left(蟺\right)脳\sin 鈦�(蟺脳(鈭�3\left)\right)$

$f_y(蟺,鈭�3)=蟺脳鈭�\sin 鈦�\left(3蟺\right)$

$f_y(蟺,鈭�3)=蟺脳0$

$(鈭�\sin 鈦�(3蟺)=0)$

Equation $3$

$f_y(\mathrm{蟺},-3)=0$

$f\left(x_0,y_0\right)=f(蟺,鈭�3)=\cos 鈦�(蟺脳(鈭�3\left)\right)$

$f(蟺,鈭�3)=\cos 鈦�(鈭�3蟺)$

Equation $4$

$f(\mathrm{蟺},-3)=-1$

Equation $2,3$and $4$are substituted in equation $1$.

We get

$0(x鈭�蟺)+0(y+3)=z+1$

$0+0=z+1$

Finally, the result is$z=-1$.