It doesn't sound to me like you want the derivative at all. It sounds like you want a low-pass filter. A low-pass filter simply removes the quickly changing data and leaves in it's place the longer, slower-changing trends. The most intuitive low-pass filter is a moving average filter, where you take the average of the last n-inputs, where n is determined based on the noise vs the size of the trend you are looking for. This is widely used from audio data to image processing to unemployment data (the four-week moving average unemployment figure is widely cited).
It's possible develop more efficient/selective filters using recursive techniques, if you feel that's necessary. You can use this tutorial to create a low-pass filter. It's written for audio, but it will work on most any data. It shows you how to write a bell filter, but a low-pass filter is easier.
http://blog.bjornroche.com/2012/08/basic-audio-eqs.html
