it is not undefined with polar coordinates, although multi-valued
> No. Even a vertical line has a slope in terms of an angle. The slope is defined everywhere. Likewise the slope of a straight line is defined everywhere -- in accordance with John Gabriel's New Calculus definition. > > > > > > > 2. Is the derivative a special kind of slope? > > > > This question makes a lot of sense. Slope always means the inclination of a straight line with the horizontal or vertical. Derivative always means the slope of a special kind of straight line called a tangent line. > > > > > > > 3. Do the tangent lines to any function have slopes that ever change? > > > > > > No. The slopes of tangent lines never change. > > > > > > > 4. Is an instantaneous rate of change real? > > > > No. It is a misconception by ignorant mathematicians whom John Gabriel has corrected. > > > > > After all, f(x)=5x^6 - 3x^5 + 2 has not changed in the last trillion light years or before that. It's tangent lines have always had the same predictable slopes. > > > > This argument makes perfect sense. If we plotted f(x) one trillion light years ago and we plotted it today, it would have exactly the same tangent lines whose slopes are always the same, because ALL tangent lines are straight lines.