Let $\mathrm{f\_\_x}\left(x\,y\right)$ represent the partial derivative function with respect to x, and let $\mathrm{f\_\_y}\left(x\,y\right)$ represent the partial derivative function with respect to y.

The gradient of a function $f\left(x\,y\right)$, symbolized $\nabla f\left(x\,y\right)$, is defined

$\nabla f\left(x\,y\right) \= \left(\mathrm{f\_\_x}\left(x\,y\right)\,\mathrm{f\_\_y}\left(x\,y\right)\mathit{ }\right)$

You can also evaluate the gradient at any particular values $\left(x\,y\right) \= \left(a\,b\right)$.

The tangent plane of $f\left(x\,y\right)$ at a point $\left(a\,b\right)$ is defined

$z \= f\left(a\,b\right) \+ \mathrm{f\_\_x}\left(a\,b\right)\left(x-a\right) \+ \mathrm{f\_\_y}\left(a\, b\right)\left(y-b\right)$

The gradient is often interpreted as a vector.

Let $\mathrm{\α} \= \left[\begin{array}{c}a\\ b\end{array}\right]\, \mathrm{\γ} \=\left[\begin{array}{c}x\\ y\end{array}\right]\, and \nabla f\left(\mathrm{\γ}\right) \= \left[\begin{array}{c}\mathrm{f\_\_x}\left(\mathrm{\γ}\right)\\ \mathrm{f\_\_y}\left(\mathrm{\γ}\right)\end{array}\right]$

Then, we can write the formula for the tangent plane as

$f\left(\mathrm{\α}\right) \+ \nabla f\left(\mathrm{\α}\right)\cdot \left(\mathrm{\γ}-\mathrm{\α}\right)$

using the dot product.$$