Partial Derivatives

The derivative f'(x) of a function f(x) gives the slope of the function at x. It is defined as

\begin{eqnarray} {df(x)\over dx} ≡ f'(x) :={\mathop{ lim}}_{h→0}{f(x + h) − f(x)\over h} .& & %&(3.4) \\ \end{eqnarray}

\begin{eqnarray*} f'(x)dx& & %& \\ \end{eqnarray*}

gives the change of the height of the curve f(x) (measured from the x–axis) at the point x, if we move a tiny step dx along the x–axis.

3.2.2 Derivatives for functions of two variables

For a function f(x,y) with two independent variables, in a certain point (x,y) we can define the slope in either the x– or the y–direction. These two give rise to the partial derivatives

\begin{eqnarray} {∂\over ∂x}f(x,y) :={\mathop{ lim}}_{h→0}{f(x + h,y) − f(x,y)\over h} & & %& \\ {∂\over ∂y}f(x,y) :={\mathop{ lim}}_{h→0}{f(x,y + h) − f(x,y)\over h} .& & %&(3.5) \\ \end{eqnarray}

Partial derivative {∂\over ∂x}f(x,{y}_{0})

The geometrical meaning of this is a follows: we keep y = {y}_{0} constant and consider the surface f(x,y) along the x–direction, i.e. the curve f(x,{y}_{0}) on the surface that appears through the cross–section with the plane y = {y}_{0} parallel to the x–z–plane. The partial derivative {∂\over ∂x}f(x,{y}_{0}) gives the slope of this curve at x. In other words: the partial derivative {∂\over ∂x}f(x,y) gives the slope of the surface at (x,y) in x–direction.

\begin{eqnarray*} {∂\over ∂x}f(x,y)dx& & %& \\ \end{eqnarray*}

gives the change of the height of the surface f(x,y) (measured from the x–y–plane) at the point (x,y), if we move a tiny step dx along the x–direction.

Partial derivative {∂\over ∂y}f({x}_{0},y)

The geometrical meaning of this is a follows: we keep x = {x}_{0} constant and consider the surface f(x,y) along the y–direction, i.e. the curve f({x}_{0},y) on the surface that appears through the cross–section with the plane x = {x}_{0} parallel to the y–z–plane. The partial derivative {∂\over ∂y}f({x}_{0},y) gives the slope of this curve at y. In other words: the partial derivative {∂\over ∂y}f(x,y) gives the slope of the surface at (x,y) in y–direction.

\begin{eqnarray*} {∂\over ∂y}f(x,y)dy& & %& \\ \end{eqnarray*}

gives the change of the height of the surface f(x,y) (measured from the x–y–plane) at the point (x,y), if we move a tiny step dy along the y–direction.

Total change of height (total differential)

\begin{eqnarray} df(x,y) := {∂\over ∂x}f(x,y)dx + {∂\over ∂y}f(x,y)dy& & %&(3.6) \\ \end{eqnarray}

is called the total differential of f(x,y) at the point (x,y) and gives the total change of the height of the surface f(x,y) (measured from the x–y–plane) at the point (x,y), if we move a tiny step dx along the x–direction and a tiny step dy along the y–direction.

How to calculate partial derivatives

Examples

\begin{eqnarray*} f(x,y)& =& {x}^{2} + {y}^{2} ⇝ {∂\over ∂x}f(x,y) = 2x,\quad {∂\over ∂y}f(x,y) = 2y %& \\ f(x,y)& =& {x}^{2}{y}^{3} ⇝ {∂\over ∂x}f(x,y) = 2x{y}^{3},\quad {∂\over ∂y}f(x,y) = {x}^{2}3{y}^{2}. %& \\ f(x,y)& =& {e}^{−xy} ⇝ {∂\over ∂x}f(x,y) = −y{e}^{−xy},\quad {∂\over ∂y}f(x,y) = −x{e}^{−xy}.%& \\ \end{eqnarray*}

3.2.3 Higher Derivatives, Notation

Higher partial derivatives are easily defined: The second partial derivative {{∂}^{2}\over ∂{x}^{2}} f(x,y) is the partial derivative with respect to x of the partial derivative {∂\over ∂x}f(x,y), etc. To simplify the notation, one often defines

\begin{eqnarray}{ f}_{x} ≡ {∂\over ∂x}f(x,y),\quad {f}_{xx}& ≡& {{∂}^{2}\over ∂{x}^{2}}f(x,y) := {∂\over ∂x} {∂\over ∂x}f(x,y) %& \\ {f}_{y} ≡ {∂\over ∂y}f(x,y),\quad {f}_{yy}& ≡& {{∂}^{2}\over ∂{y}^{2}}f(x,y) := {∂\over ∂y} {∂\over ∂y}f(x,y) %& \\ {f}_{xy}& ≡& {{∂}^{2}\over ∂x∂y}f(x,y) := {∂\over ∂x} {∂\over ∂y}f(x,y) %& \\ {f}_{yx}& ≡& {{∂}^{2}\over ∂y∂x}f(x,y) := {∂\over ∂y} {∂\over ∂x}f(x,y).%&(3.7) \\ \end{eqnarray}

Examples of Higher Partial Derivatives

\begin{eqnarray*} f(x,y)& =& {x}^{2} + {y}^{2} ⇝ {f}_{ x}(x,y) = 2x,\quad {f}_{y}(x,y) = 2y %& \\ {f}_{xx}(x,y)& =& {f}_{yy}(x,y) = 2,\quad {f}_{xy}(x,y) = {f}_{yx}(x,y) = 0.%& \\ \end{eqnarray*}

3.2.4 Minima, Maxima and Saddle points

Extrema (stationary points) occur when all partial derivatives are zero. Types: minima, maxima and saddlepoints (minimum in one direction, maximum in another).

3.2 Partial Derivatives

3.2.1 Reminder: Derivative of a function of one variable