Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/215



Since $$x$$ and $$y$$ are independent in

$$x$$ may be supposed to vary while $$y$$ remains constant, or the reverse.

The derivative of $$z$$ with respect to $$x$$ when $$x$$ varies and $$y$$ remains constant is called the partial derivative of $$z $$ with respect to $$x$$, and is denoted by the symbol $$\tfrac{\partial z}{\partial z}$$ We may then write

Similarly, when $$x$$ remains constant and $$y$$ varies, the partial derivative of z with respect to $$y$$ is

In order to avoid confusion the round $$\partial$$ has been generally adopted to indicate partial differentiation. Other notations; however, which are in use are

Our notation may be extended to a function of any number of independent variables. Thus, if

then we have the three partial derivatives

Find the partial derivatives of $$z = ax^2 + 2 bxy + cy^2.$$




 * Solution.
 * $$\tfrac{\partial z}{\partial x} = 2 ax + 2 by$$, treating $$y$$ as a constant,
 * $$\tfrac{\partial z}{\partial y} = 2 bx + cy$$, treating $$x$$ as a constant.
 * }
 * $$\tfrac{\partial z}{\partial y} = 2 bx + cy$$, treating $$x$$ as a constant.
 * }