Calculate exponential growth when rate r is changing

[Vodkacannon]

The formula f(t) = I(1 + r)^t gives the value of something after it has grown for t years, at rate r, with an initial value I.

What if the growth rate r is continuously increasing?
What is the formula now?
Does it relate somehow to x^x?

 

[haruspex]

It depends on how it increases with t. Is it linear with t? Geometric? Logarithmic? ...

 

[HallsofIvy]

It is, of course, $f(t)= I(1+ r(t))^t$. And, yes, that has similar properties to $x^x$, especially if r(t) is linear.