suppose that the continous function f R2 to R has a tangent

suppose that the continous function f: R2 to R has a tangent plane at the point (x1,y1,f(x1,y1)). prove that the function f: R2 to R has a directional derivative in all direction at point (x1,y1)

Solution

The function is z = f(x,y)

The tangent plane will have equation as

Z-f(x1,y1) = fx (x-x1) +fy(y-y1)

Thus fx, fy are the derivatives (partial of f)

Directional derivative along the point (x1,y1) is

fx (x-x1)+fy(y-y1)

Thus proved