Directional Derivatives in Any Direction

Directional derivatives measure how a surface changes in any chosen direction. In multivariable calculus, partial derivatives show change along one coordinate at a time. For a function of two variables, one partial derivative gives the slope in the x-direction, and the other gives the slope in the y-direction.

In a multivariable function, partial derivatives measure the slope solely either along the direction parallel to the x axis or along the direction parallel to the y axis.

However, real-world motion, such as a hiker navigating a mountain, is rarely restricted to these fixed directions. To find the slope in any arbitrary direct...