The derivative of a function gives the slope of the function at . It is defined as
Change of height: the quantity
gives the change of the height of the curve (measured from the –axis) at the point , if we move a tiny step along the –axis.
For a function with two independent variables, in a certain point we can define the slope in either the – or the –direction. These two give rise to the partial derivatives
The geometrical meaning of this is a follows: we keep constant and consider the surface along the –direction, i.e. the curve on the surface that appears through the cross–section with the plane parallel to the – –plane. The partial derivative gives the slope of this curve at . In other words: the partial derivative gives the slope of the surface at in –direction.
Change of height: the quantity
gives the change of the height of the surface (measured from the – –plane) at the point , if we move a tiny step along the –direction.
The geometrical meaning of this is a follows: we keep constant and consider the surface along the –direction, i.e. the curve on the surface that appears through the cross–section with the plane parallel to the – –plane. The partial derivative gives the slope of this curve at . In other words: the partial derivative gives the slope of the surface at in –direction.
Change of height: the quantity
gives the change of the height of the surface (measured from the – –plane) at the point , if we move a tiny step along the –direction.
The quantity
is called the total differential of at the point and gives the total change of the height of the surface (measured from the – –plane) at the point , if we move a tiny step along the –direction and a tiny step along the –direction.
This is very simple:
Higher partial derivatives are easily defined: The second partial derivative is the partial derivative with respect to of the partial derivative , etc. To simplify the notation, one often defines
Extrema (stationary points) occur when all partial derivatives are zero. Types: minima, maxima and saddlepoints (minimum in one direction, maximum in another).