The notion of a derivative has numerous interpretations and applications. A well-known geometric interpretation is that of a slope, or more generally that of a linear approximation to a mapping between linear spaces (see here). Another useful interpretation comes from physics and is based on the idea of related rates. This second point of view is quite general, and sheds light on the definition of the derivative of a manifold mapping (the latter is described in the pushforward entry).

Consider two physical quantities $x$ and $y$ that are somehow coupled. For example:

• the quantities $x$ and $y$ could be the coordinates of a point as it moves along the unit circle;
• the quantity $x$ could be the radius of a sphere and $y$ the sphere's surface area;
• the quantity $x$ could be the horizontal position of a point on a given curve and $y$ the distance traversed by that point as it moves from some fixed starting position;
• the quantity $x$ could be depth of water in a conical tank and $y$ the rate at which the water flows out the bottom.

It does not matter how we vary the two quantities, the ratio of the rates of change depends only on the values of $x$ and $y$ . This ratio is, of course, the derivative of the function that maps the values of $x$ to the values of $y$ . Letting $\dot{x}, \dot{y}$ denote the rates of change of the two quantities, we describe this conception of the derivative as $$ \frac{dy}{dx} = \frac{\dot{y}}{\dot{x}},$$ or equivalently as \begin{equation} \label{eq:deriv} \dot{y} = \frac{dy}{dx}\, \dot{x}. \end{equation} Next, let us generalize the discussion and suppose that the two quantities $\bx$ and $\by$ represent physical states with multiple degrees of freedom. For example, $\bx$ could be a point on the earth's surface, and $\by$ the position of a point 1 kilometer to the north of $\bx$ . Again, the dependence of $\by$ and $\bx$ is, in general, non-linear, but the rate of change of $\by$ does have a linear dependence on the rate of change of $\bx$ . We would like to say that the derivative is precisely this linear relation, but we must first contend with the following complication. The rates of change are no longer scalars, but rather velocity vectors, and therefore the derivative must be regarded as a linear transformation that changes one vector into another.

In order to formalize this generalized notion of the derivative we must consider $\bx$ and $\by$ to be points on manifolds $X$ and $Y$ , and the relation between them a manifold mapping $\phi:X\rightarrow Y$ . A varying $\bx$ is formally described by a parameterized curve $$\gamma:I\rightarrow X,\quad I\subset\reals.$$ The corresponding velocities take their value in the tangent spaces of $X$ : $$\gamma'(t) \in \rT_{\gamma(t)} X.$$ The ``coupling'' of the two quantities is described by the composition $$\phi\circ\gamma:I\rightarrow Y.$$ The derivative of $\phi$ at any given $\bx\in X$ is a linear mapping $$\phi_*(\bx): \rT_{\bx} X \rightarrow \rT_{\phi(\bx)} Y,$$ called the pushforward of $\phi$ at $\bx$ , with the property that for every trajectory $\gamma$ passing through $\bx$ at time $t$ , we have $$(\phi\circ\gamma)'(t) = \phi_*(\bx)\gamma'(t).$$ The above is the multi-dimensional and coordinate-free generalization of the related rates relation ().

All of the above has a perfectly rigorous presentation in terms of manifold theory. The approach of the present entry is more informal; our ambition was merely to motivate the notion of a derivative by describing it as a linear transformation between velocity vectors.