91Ó°ÊÓ

The given is the function is $f(x,y)$is differentiable at the point $(a,b)$

The objective is to find the equation of the plane tangent

Let $f(x,y)$be defined as a function of two variables on an open set including the point $(a,b)$and let $\mathrm{Δ}z=f(a+\mathrm{Δ}x,b+\mathrm{Δ}y)-f(a,b)$. The function $f$is said to be differentiable if $f_x(a,b)and{f}_y(a,b)$both exists and

$\mathrm{Δ}z=f_x(a,b)\mathrm{Δ}x+f_y(a,b)\mathrm{Δ}y+{\mathrm{Î\mu }}_1\mathrm{Δ}x+{\mathrm{Î\mu }}_2\mathrm{Δ}y$

where ${\mathrm{Î\mu }}_1$and ${\mathrm{Î\mu }}_2$are functions of $\mathrm{Δ}x$and $\mathrm{Δ}y$, and they're both zero when $(\mathrm{Δ}x,\mathrm{Δ}y)→(0,0)$.

Substitute $\mathrm{Δ}x=x-a,\mathrm{Δ}y=y-b$and $\mathrm{Δ}z=z-f(a,b)$in $\left(1\right)$and eliminate ${\mathrm{Î\mu }}_1∆xand{\mathrm{Î\mu }}_2∆y$terms.

$z-f(a,b)=f_x(a,b)(x-a)+f_y(a,b)(y-b)$

$=∇f_{+}(a,b)·⟨(x,y)-(a,b)âŸ\copyright$

As a result, the tangent plane to the surface equation $f(x,y)$ at $(a,b)$ is