Here is the whole process again in a simple animation.
We start with a function with two inputs and one output. In this case the function f(x, y) = x2 + y2 - ¼xy + x - y +1
We pick a point on the function, in this case (10, 20).
We can define two partial derivatives. One is the derivative of f as x varies and y is kept fixed.
The other is the derivative of f as y varies and x is kept fixed.
Both of these are functions of one variable, so we can apply what we know from univariate calculus to work out the derivatives. At our point (10, 20), this gives us a tangent line touching the red function and a tangent line touching the blue function.
Since these lines cross, they lie in a shared hyperplane. That is the plane that (in most cases) just touches but does not cross f. Like the tangent line, the tangent hyperplane functions as a locally linear approximation of f: in a small enighborhood around the point (10, 20), it behaves as much like f as any linear function can.
+Here is the whole process again in a simple animation.
We start with a function with two inputs and one output. In this case the function f(x, y) = x2 + y2 - ¼xy + x - y +1
We pick a point on the function, in this case (10, 20).
We can define two partial derivatives. One is the derivative of f as x varies and y is kept fixed.
The other is the derivative of f as y varies and x is kept fixed.
Both of these are functions of one variable, so we can apply what we know from univariate calculus to work out the derivatives. At our point (10, 20), this gives us a tangent line touching the red function and a tangent line touching the blue function.
Since these lines cross, they lie in a shared hyperplane. That is the plane that (in most cases) just touches but does not cross f. Like the tangent line, the tangent hyperplane functions as a locally linear approximation of f: in a small neighborhood around the point (10, 20), it behaves as much like f as any linear function can.