Measuring the slope of an Nd-curve

In order to measure the slope of a vector or a multi-dimensional surface, we will introduce the idea of partial derivatives, which are simply derivatives with respect to a variable, with all the other variables held as constants. So, our solution is a vector of dimension k, where k is the number of variables that our function takes. In this case, we have x and y. Each respective position in the vector that we solve is a derivative with respect to the corresponding function's positional variable.

From a conceptual level, what we're doing is we're holding one of the variables still and changing the other variables around it to see how the slope changes. Our denominator's notation indicates which variable we're measuring the slope with, with respect to that point. So, in this case, the first position, d(x), is showing that we're taking the partial derivative of function f with respect to x, where we hold y constant. And then, likewise, in the second one, we're taking the derivative of function f with respect to y, holding x constant. So, what we get in the end is called a gradient, which is a super keyword. It is simply just a vector of partial derivatives: