I've seen several answers but non of them do not answer what I am looking for, therefore I am asking here again,
According to my understanding,
Slope, $\frac{\partial f}{\partial x}$, is generally applicable when only 2 variables are in consideration. The slope is the tangent or the derivative to the function's curve that connects the 2 variables, i.e., the measure of the rate of change of a function f(x) with respect to the x.
Gradient is the transpose derivatives or just the vector of partial derivatives.
Consider $f(x,y) = y-x$
Gradient of this $\nabla f(x,y) = \frac{\partial f(x,y)}{\partial x}i + \frac{\partial f(x,y)}{\partial y}j = -i+j$
Question:
Since the slope or tangent line and gradient are perpendicular to each other is there a way to prove this?