Exponential Derivative Word Problem I am having problem with a world problem derivative application question. 
The number of parasites in the blood after $h$ hours medication is taken is given by the function $p = \dfrac{200}{0.9+0.1e^{2h}}$. 
I am to find out after how many hours is the number of parasites decreasing the fastest?
How should I solve this question? What I had in mind was to find the first derivative and locate it's zeros. Would this be a good approach?
 A: So you need to find when its decreasing the fastest. In other words, you need to find when the derivative is negative and  has the largest negative value. (Decreasing means its derivative is negative)
So first perform the derivative to get a new function: the rate of change.
Now we need to find when the rate of change has the largest negative value. Picture two hills. The smallest point between two hills is the ground in the middle. That point is where the ground becomes flat. A flat graph means we are not changing at all so we need to find where the change is 0, in other words when the derivative is 0
So now perform the derivative of the derivative and find the 0's of that function
Finally, the top of the "hills" are flat to. How do we determine the difference between the top and the bottom (we want to find the smallest, the bottom). To do this we need to find a point where the function goes from decreasing to increasing (the bottom slope). So using the second derivative, plug in points in between each x coordinate of the zeroes you've found and find out if that value is positive or negative. Example: you've found zeroes of 5, 9 and 16. Some good x values to plug into the function are 3, 7, 11, and 17.
Now list all zeroes where a test point was negative before and a test point was positive after. These are your minimum values.
Now take the zeroes that you've found (which were points on bottoms of "hills") and plug the x coordinate to the first derivative. Pick the value that was the largest negative number and that's your answer.
It's a lengthy answer but I promise you its a lot faster and simpler once you've practiced.
