91影视

Given Information$f(x,y)=\mathrm{Sin}鈦�\left(xy\right)$at position $P=\left(x_0,y_0\right)=\left(2,\frac{蟺}{2}\right)$

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 $1$

$f_x\left(2,\frac{蟺}{2}\right)(x鈭�2)+f_y\left(2,\frac{蟺}{2}\right)\left(y鈭�\frac{蟺}{2}\right)=z鈭�f\left(2,\frac{蟺}{2}\right)$

Then,

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

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

localid="1650520586645" $\begin{array}{r}{f}_x\left(x_0,y_0\right)={\left.\cos 鈦�\left(xy\right)脳y\right|}_{\left(x_0,y_0\right)}\\ f_x\left(x_0,y_0\right)={\left.y脳\cos 鈦�\left(xy\right)\right|}_{\left(x_0,x_0\right)}\end{array}$

$f_x\left(2,\frac{蟺}{2}\right)=y脳\cos 鈦�\left(xy\right)\left(2,\frac{蟺}{2}\right)$

$f_x\left(2,\frac{蟺}{2}\right)=\frac{蟺}{2}\cos 鈦�\left(2脳\frac{蟺}{2}\right)$

$(鈭�\cos 鈦�(蟺)=鈭�1)$

$f_x\left(2,\frac{蟺}{2}\right)=鈭�\frac{蟺}{2}$

Equation $2$

$f_x\left(2,\frac{蟺}{2}\right)=鈭�\frac{蟺}{2}$

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

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

$\begin{array}{r}{f}_y\left(x_0,y_0\right)={\left.\cos 鈦�\left(xy\right)脳x\right|}_{\left(x_0,y_6\right)}\\ \end{array}$

$f_y\left(x_0,y_0\right)={\left.x脳\cos 鈦�\left(xy\right)\right|}_{\left(x_0,y_0\right)}$

$f_y\left(2,\frac{蟺}{2}\right)=x脳\cos 鈦�\left(xy\right)鈭�\left(2,\frac{蟺}{2}\right)$

$f_y\left(2,\frac{蟺}{2}\right)=2\cos 鈦�\left(2脳\frac{蟺}{2}\right)$

$f_y\left(2,\frac{蟺}{2}\right)=鈭�2$

$(鈭�\cos 鈦�(蟺)=鈭�1)$

Equation $3$

$f_y\left(2,\frac{蟺}{2}\right)=鈭�2$

$f\left(x_0,y_0\right)=f\left(2,\frac{蟺}{2}\right)=\sin 鈦�\left(2脳\frac{蟺}{2}\right)$

$f\left(2,\frac{蟺}{2}\right)=\sin 鈦�\left(蟺\right)$

Equation $4$

$f\left(2,\frac{蟺}{2}\right)=0$

Equations $2,3$and $4$are substituted by equation $1$, we get,

$鈭�\frac{蟺}{2}(x鈭�2)鈭�2\left(y鈭�\frac{蟺}{2}\right)=z鈭�0$

$鈭�\frac{蟺}{2}x+\frac{蟺}{2}脳2鈭�2y+2脳\frac{蟺}{2}=z$

$鈭�\frac{蟺}{2}x鈭�2y鈭�z=鈭�蟺鈭�蟺$

$鈭�\frac{蟺}{2}x鈭�2y鈭�z=鈭�2蟺$

$2$is multiplied by two sides,

$蟺x+4y+2z=4蟺$

Finally, the result for expression $蟺x+4y+2z=4蟺$