Higher Derivatives

Note that a partial derivative is itself a function of two variables, and so further partial derivatives can be calculated. We write

$${\frac{{\partial}}{{\partial x}}}$$$${\frac{{\partial f}}{{\partial x}}}$$ = $${\frac{{\partial^{2}f}}{{\partial x^{2}}}}$$,        $${\frac{{\partial}}{{\partial x}}}$$$${\frac{{\partial f}}{{\partial y}}}$$ = $${\frac{{\partial^{2}f}}{{ \partial{\if 11x\else x^{1}\fi} \partial{\if 11y\else y^{1}\fi} }}}$$,        $${\frac{{\partial}}{{\partial y}}}$$$${\frac{{\partial f}}{{\partial x}}}$$ = $${\frac{{\partial^{2}f}}{{ \partial{\if 11y\else y^{1}\fi} \partial{\if 11x\else x^{1}\fi} }}}$$,        $${\frac{{\partial}}{{\partial y}}}$$$${\frac{{\partial f}}{{\partial y}}}$$ = $${\frac{{\partial^{2}f}}{{\partial y^{2}}}}$$.

This notation generalises to more than two variables, and to more than two derivatives in the way you would expect. There is a complication that does not occur when dealing with functions of a single variable; there are four derivatives of second order, as follows:

$${\frac{{\partial^{2}f}}{{\partial x^{2}}}}$$,        $${\frac{{\partial^{2}f}}{{ \partial{\if 11x\else x^{1}\fi} \partial{\if 11y\else y^{1}\fi} }}}$$,        $${\frac{{\partial^{2}f}}{{ \partial{\if 11y\else y^{1}\fi} \partial{\if 11x\else x^{1}\fi} }}}$$    and    $${\frac{{\partial^{2}f}}{{\partial y^{2}}}}$$.

Fortunately, when f has mild restrictions, the order in which the differentiation is done doesn't matter.

Proposition 8.12   Assume that all second order derivatives of f exist and are continuous. Then the mixed second order partial derivatives of f are equal. i.e.

$${\frac{{\partial^{2}f}}{{ \partial{\if 11x\else x^{1}\fi} \partial{\if 11y\else y^{1}\fi} }}}$$ = $${\frac{{\partial^{2}f}}{{ \partial{\if 11y\else y^{1}\fi} \partial{\if 11x\else x^{1}\fi} }}}$$.

Example 8.13   Suppose that f (x, y) is written in terms of u and v where x  =  u + log v and y  =  u - log v. Show that, with the usual convention,

$${\frac{{\partial^{2}f}}{{\partial u^{2}}}}$$ = $${\frac{{\partial^{2}f}}{{\partial x^{2}}}}$$ + 2$${\frac{{\partial^{2}f}}{{ \partial{\if 11x\else x^{1}\fi} \partial{\if 11y\else y^{1}\fi} }}}$$ + $${\frac{{\partial^{2}f}}{{\partial y^{2}}}}$$

and

v²$${\frac{{\partial^{2}f}}{{\partial v^{2}}}}$$ = $${\frac{{\partial f}}{{\partial y}}}$$ - $${\frac{{\partial f}}{{\partial x}}}$$ + $${\frac{{\partial^{2}f}}{{\partial x^{2}}}}$$ + 2$${\frac{{\partial^{2}f}}{{ \partial{\if 11x\else x^{1}\fi} \partial{\if 11y\else y^{1}\fi} }}}$$ + $${\frac{{\partial^{2}f}}{{\partial y^{2}}}}$$

You may assume that all second order derivatives of f exist and are continuous.

Solution. Using the chain rule, we have

$${\frac{{\partial f}}{{\partial u}}}$$ = $${\frac{{\partial f}}{{\partial x}}}$$$${\frac{{\partial x}}{{\partial u}}}$$ + $${\frac{{\partial f}}{{\partial y}}}$$$${\frac{{\partial y}}{{\partial u}}}$$ = $${\frac{{\partial f}}{{\partial x}}}$$ + $${\frac{{\partial f}}{{\partial y}}}$$    and    $${\frac{{\partial f}}{{\partial v}}}$$ = $${\frac{{\partial f}}{{\partial x}}}$$$${\frac{{\partial x}}{{\partial v}}}$$ + $${\frac{{\partial f}}{{\partial y}}}$$$${\frac{{\partial y}}{{\partial v}}}$$ = $${\frac{{1 }}{{ v}}}$$$${\frac{{\partial f}}{{\partial x}}}$$ - $${\frac{{1 }}{{ v}}}$$$${\frac{{\partial f}}{{\partial y}}}$$.

Thus using both these and their operator form, we have

$${\frac{{\partial^{2}f}}{{\partial u^{2}}}} {\frac{{\partial}}{{\partial u}}} \left(\vphantom{\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} }\right. {\frac{{\partial f}}{{\partial x}}} {\frac{{\partial f}}{{\partial y}}} \left.\vphantom{\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} }\right) {\frac{{\partial}}{{\partial x}}} {\frac{{\partial}}{{\partial y}}} \left(\vphantom{\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} }\right. {\frac{{\partial f}}{{\partial x}}} {\frac{{\partial f}}{{\partial y}}} \left.\vphantom{\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} }\right) {\frac{{\partial^{2}f}}{{\partial x^{2}}}} {\frac{{\partial^{2}f}}{{ \partial{\if 11x\else x^{1}\fi} \partial{\if 11y\else y^{1}\fi} }}} {\frac{{\partial^{2}f}}{{ \partial{\if 11y\else y^{1}\fi} \partial{\if 11x\else x^{1}\fi} }}} {\frac{{\partial^{2}f}}{{\partial y^{2}}}}$$ =

$${\frac{{\partial^{2}f}}{{\partial v^{2}}}} {\frac{{\partial}}{{\partial v}}} \left(\vphantom{\frac{1 }{ v}\frac{\partial f}{\partial x} -\frac{1}{ v} \frac{\partial f}{\partial y}}\right. {\frac{{1 }}{{ v}}} {\frac{{\partial f}}{{\partial x}}} {\frac{{1 }}{{ v}}} {\frac{{\partial f}}{{\partial y}}} \left.\vphantom{\frac{1 }{ v}\frac{\partial f}{\partial x} -\frac{1}{ v} \frac{\partial f}{\partial y}}\right) {\frac{{1 }}{{ v^2}}} {\frac{{\partial f}}{{\partial x}}} {\frac{{1 }}{{ v}}} {\frac{{\partial}}{{\partial v}}} \left(\vphantom{ \frac{\partial f}{\partial x} }\right. {\frac{{\partial f}}{{\partial x}}} \left.\vphantom{ \frac{\partial f}{\partial x} }\right) {\frac{{1 }}{{ v^2}}} {\frac{{\partial f}}{{\partial y}}} {\frac{{1 }}{{ v}}} {\frac{{\partial}}{{\partial v}}} \left(\vphantom{ \frac{\partial f}{\partial y} }\right. {\frac{{\partial f}}{{\partial y}}} \left.\vphantom{ \frac{\partial f}{\partial y} }\right)$$ =

$${\frac{{1 }}{{ v^2}}} {\frac{{\partial f}}{{\partial x}}} {\frac{{1 }}{{ v}}} \left(\vphantom{\frac{1 }{ v}\frac{\partial^{2}f}{\partial x^{2}}... ...}f}{ \partial{\if 11y\else y^{1}\fi} \partial{\if 11x\else x^{1}\fi} } }\right. {\frac{{1 }}{{ v}}} {\frac{{\partial^{2}f}}{{\partial x^{2}}}} {\frac{{1 }}{{ v}}} {\frac{{\partial^{2}f}}{{ \partial{\if 11y\else y^{1}\fi} \partial{\if 11x\else x^{1}\fi} }}} \left.\vphantom{\frac{1 }{ v}\frac{\partial^{2}f}{\partial x^{2}}... ...}f}{ \partial{\if 11y\else y^{1}\fi} \partial{\if 11x\else x^{1}\fi} } }\right) {\frac{{1 }}{{ v^2}}} {\frac{{\partial f}}{{\partial y}}} {\frac{{1 }}{{ v}}} \left(\vphantom{\frac{1 }{ v} \frac{\partial^{2}f}{ \partial{\if ... ...1y\else y^{1}\fi} } -\frac{1}{ v} \frac{\partial^{2}f}{\partial y^{2}} }\right. {\frac{{1 }}{{ v}}} {\frac{{\partial^{2}f}}{{ \partial{\if 11x\else x^{1}\fi} \partial{\if 11y\else y^{1}\fi} }}} {\frac{{1 }}{{ v}}} {\frac{{\partial^{2}f}}{{\partial y^{2}}}} \left.\vphantom{\frac{1 }{ v} \frac{\partial^{2}f}{ \partial{\if ... ...1y\else y^{1}\fi} } -\frac{1}{ v} \frac{\partial^{2}f}{\partial y^{2}} }\right)$$ =

$${\frac{{1 }}{{ v^2}}} \left(\vphantom{ \frac{\partial f}{\partial y} - \frac{\partial f... ...artial{\if 11y\else y^{1}\fi} } + \frac{\partial^{2}f}{\partial y^{2}} }\right. {\frac{{\partial f}}{{\partial y}}} {\frac{{\partial f}}{{\partial x}}} {\frac{{\partial^{2}f}}{{\partial x^{2}}}} {\frac{{\partial^{2}f}}{{ \partial{\if 11x\else x^{1}\fi} \partial{\if 11y\else y^{1}\fi} }}} {\frac{{\partial^{2}f}}{{\partial y^{2}}}} \left.\vphantom{ \frac{\partial f}{\partial y} - \frac{\partial f... ...artial{\if 11y\else y^{1}\fi} } + \frac{\partial^{2}f}{\partial y^{2}} }\right)$$ =