We all know a derivative of a function in Cartesian coordinates can be represented (or is defined by) by the slope of a line tangent to the graph. I know that concepts of math are often extended beyond the realm of intuitivity, but I was wondering what the derivative of a polar curve represents, a change in the radius over a change in angle. I'm not convinced it's the tangent line to the graph, because the angle is not a linear measurement and doesn't have units. Does anyone have any intuition for this? Thanks.
Derivative of polar curve
calculus
Related Solutions
Hint: $$9r\sin\theta+19r\cos\theta=39\ .$$
The magic word is "rate of change". The slope is the rate of change, so the slope is simply "how much the function grows when you move right". So if you plot the integral curve (which geometrically can be interpreted as an area under some other curve), then the rate of change is "how much the area grows when you expand it to the right", which is exactly the value of the original function.
Sketch:
On the left, you have a curve of the original function $f(x)$, and the approximate area under it, shaded as red rectangles. The curve of the integral $\int f(x)dx$ is obtained by "adding" the areas together (right figure, the $y$ coordinate measures the total area of the curve in the left figure). The derivative of the right curve is the slope (dashed lines across the rectangles), and obviously, the slope of a rectangle with unit width is exactly its height, and the height of the rectangle brings you back to the left figure and the value $f(x)$.
Best Answer
Well, you seem to have hit the nail on the head!
In a Cartesian coordinates, it makes sense that $\displaystyle \frac{\Delta y}{\Delta x}$ would help create a line, because a line is compatible with the horizontal and vertical components of the xy grid.
However, in polar coordinates, a tangent line doesn't really tie well with $r$ and $\theta$ like that, especially when it doesn't pass through the origin.
So how do you find the tangent line to a polar curve, well sorry to let you down, but you'll have to parameterize it first and then go from there. Since tangent lines are using Cartesian bases.