How do you find the equation of a tangent plane to the graph of a function f(x,y)? This is the multivariable analog of finding the equation of a tangent line to the single variable function f(x). At each point (if f is differentiable) the tangent plane is a linear approximation. To compute this we use the vector equation of a plane with respect to the normal vector. The components of the normal vector come from the partial derivatives so if you can find partial derivatives you can write down the equation of the tangent plane.
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