How to use the Chain Rule for Derivatives

Every so often in physics we come across a slightly more complex function for a problem than what we are used to, and then need to find a derivative of that more complex function.  By more complex I am referring to the case of 'compound functions,' which I define as an outside function operating on an inside function.  A classic example happens with our solution to simple harmonic motion problems, where the position of a mass on a spring is defined as x(t) = Asin(wt + phi) or Acos(wt + phi), where A is amplitude, w is the angular frequency, and phi is a phase angle; these three are just constants. Here, we have an 'outside' function, sine or cosine, operating on an inside function, (wt + phi).  How would you find the derivative of this compound function, which is necessary in physics if we want to find velocity and acceleration?

The answer is the Chain Rule.  This says: the derivative is just the derivative of the outside function times the derivative of the inside function.

Check out a few examples in this video, and I suspect you will catch on within a few minutes.  I hope this helps.