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In a basic mathematics for economists course one is exposed to the concept of directional derviative. Recall that a directional derivative is defined as:
$$\nabla f \frac{\mathbb{v}}{\|\mathbb{v}\|}$$
I know that it mathematically tells you the direction the function is increasing on a two or three dimensional graph, however I am yet to see an application of such a formula in any economics problems.
What are some applications of the directional derivative in Economics?
Recall, that the directional derivative of a function is defined as the the rate at which the function $f(x,y,z)$ at a given point $(x_0, y_0, z_0)$ changes in direction of the vector $v$, where $||v||$ is the norm of $v$1.
Applications of the directional derivative can be used in determining the rate of switching inputs in production functions, which can be very helpful in determining/forecasting switching costs for a given bundle of inputs. Though it does not tell you where the optimum points are it does tell you the rate of change (i.e. the switching costs) of moving to a different bundle.
Hope this is helpful
