Math Tutorial - Partial Derivatives

In order to understand the generalization of Newtonian mechanics to two and three dimensions, we first need to understand a new type of derivative called the partial derivative.

To motivate the discussion, let us use the chain rule to take the ordinary derivative of $f(x,y) = x^2 + y^2$ where $y$ is some function of $x$, say $y = x^3 + x$:

$$\frac{df}{dx} = 2x + 2y \frac{dy}{dx} = 2x + 2y \cdot (3x^2 + 1) .$$ (9.19)

(At this point we don't really care what form $y = y(x)$ takes.) At times we may wish to take the derivative of $f$ with respect to $x$ while ignoring the possible dependence of $y$ on $x$. A derivative of this type has special notation -- the ``$d$'' is replaced by ``$\partial$'' and the derivative is called a partial derivative. Thus, the partial derivative of $f$ with respect to $x$ is

$$\frac{\partial f}{\partial x} = 2x .$$ (9.20)

Notice that the partial derivative is actually simpler to evaluate than an ordinary derivative, because only the explicit dependence of the function on the differentiating variable need be considered -- all other variables are taken to be constant.