If the area of the circle increases at a uniform rate, show that the rate of increase of the circumference varies inversely as the radius.
My approach-
If area is increasing at a uniform rate then the area function should be $A(r)=pr+q$ where $p$ and $q$ are certain constants and $r$ is the radius of the circle.
We can say that- $$A(r)=\int_0^r C(r)dr$$where $C(r)$ is the circumference as a function of $r$.
Now differentiating both sides with respect to $r$- $$A'(r)=\frac{d}{dr}(\int_0^r C(r)dr)$$
Now, using Newton-Leibniz formula- $$p=C(r)$$
Circumference turns out to be a constant quantity! Where did I go wrong?
I think you assumed that the area is linear with $r$- but the relationship between $A$ and $r$ is fixed: $A= \pi r^2$. What the question means is a constant $A'(t)$ ($A$ is linear with time). If you use the above $A(r)$, you'll get the desired result.