So I was given the following prompt:
"A polar curve is given by the equation $r=\frac{100}{\theta^2+1}$ for $\theta \ge0$. What is the instantaneous rate of change of $r$ with respect to $\theta$ when $\theta=2$?"
I guess I'm a bit confused about what $\frac{dy}{dx}$ is going to look like here. I thought I was able to find the derivative of the equation by plugging the function for $r$ into the equations of $y=r\sin\theta$ and $x=r\cos\theta$, but after finding the value of the derivative that I ended at, at $\theta=2$, I'm getting a weird answer. Any help would be appreciated!
Best Answer
Hint.
All you need to find is $$\frac{dr}{d\theta}\bigg|_{\theta=2}$$
There are no Cartesian coordinates in this problem.
In general, if the quantity $z$ is a function of another quantity $w$ where $z=f(w)$ and $f$ is differentiable, then the "instantaneous rate of change of $z$ with respect to $w$ when $w=a$" is given by $f'(a)$.