Im am trying to understand the idea of derivatives and how they relate to the real world.
I understand if i have function, in pkysics first derivative is the velocity, and the second derivative is the acceleration. My question is are there any other physical or theoretical applications of the derivatives that are similar. I ask also if i can apply this to probability, and in particular what is the relationship between the probability densitiy function and cumulative distribution
The probability density function is the first derivative of the cumulative distribution. So that gives you an application of derivatives in probability.
If you aren't familiar with Taylor series yet, that is another very important application of derivatives (all derivatives of every order) so that you can approximate a smooth function by a polynomial, to arbitrary accuracy as you let the degree of the polynomial increase.