I'm working my way through all the Khan Academy topics in the practice section and I've come as far as 'special derivatives' and 'chain rule'. The special derivatives consist of identifying the derivatives of sin, cos, tan, ln and e. I have memorized the derivative of these functions but I do not understand why they work (cannot find a good proof).
In the chain rule and beyond, almost every problem has these special derivatives in them and I am finding it really hard to find good information containing proofs for these.
For example I memorized that the derivative of ln is 1/x and also I memorized that the derivative of tan(x) is sec^2(x). Then I got a problem which asked me to derive ln*tan(x) which somehow started going in the direction of 1/tan(x) and I'm clueless.
Please help me find some good stuff on this as I've been stuck here for a long time and I was progressing extremely fast in math before this!