> If you start to really dig into the underpinnings of the topic, Let's cover the most important part of partial derivatives, the geometric intuition. Imagine

> If you start to really dig into the underpinnings of the topic, Let's cover the most important part of partial derivatives, the geometric intuition. Imagine the Smoky Mountains of east Tennessee, that is, smooth, rolling hills. Now to represent this landscape in math, let R be the set of real numbers, R^2 pairs of real numbers, that is, the coordinates of points in the plane with orthogonal axes X and Y, and let f: R^2 --> R, that is, f is a function of two variables, say, x and y, that is, the pair (x,y) in R^2, and the value f(x,y) is the height of the mountains above point (x,y), that is, a plane under the mountains. Then at a point (x, y) the partial derivative of f(x,y) with respect to x is just the slope as in ordinary derivative of the mountain at point (x,y) in the direction of changing x. So, if the X axis runs east and west, the partial derivative of f(x,y) with respect to x is the slope of the mountain at (x,y) in the east-west direction. So the partial derivative is just like the derivative of a function of one variable, that is, a slope, except is for just one variable, say, x, with the other variable(s) y held constant. So, if f(x,y) = 3xy + 2x - y then the partial derivative of f(x,y) with respect to x is just D_x f(x,y) = 3y + 2 and the partial derivative of f(x,y) with respect to y is D_y f(x,y) = 3x - 1 These partial derivatives are important in vector analysis and, thus, Maxwell's equations, electro-magnetism, fluid flow, optimization, etc.