Math Is Fun Forum

I've had a good look at all the existing topics based around this rather annoying subject, but nothing seems to satisfyingly relate to what I'm questioning. I don't quite understand how special functions react to differentiation, well I suppose I do, if basic differentiation. I'm just really having problems with questions involving the differentiation of multiple functions that are multiplied or squared.

There's a heck of a lot of questions I'm having problems with, and I'd much rather learn how to solve them as oppose to watching someone else work through it for me. So, I was thinking, would it be possible if someone could fill out a list of how the functions are differentiated, maybe if I were to provide a structure? At least, I know this:

f(x)          f'(x)   

sin(x)          cos(x)
cos(x)      -sin(x)
tan(x)         sec²(x)
sec(x)         sec(x)tan(x)
cosec(x)   -cosec(x)cot(x)
cot(x)          -cosec²(x)
ln(x)           1/x
e^x          e^x

I know, it's not much at all, but I'm just quite unsure of the following:

f(x)              f'(x)   

sin(ax)         
cos(ax)       
tan(ax)         
sec(ax)         
cosec(ax)     
cot(ax)         
ln(ax)             
e^(ax)         

f(x)            f'(x)   

sin²(x)       
cos²(x)       
tan²(x)       
sec²(x)         
cosec²(x)   
cot²(x)             
ln²(x)           

f(x)               f'(x)   

a x sin(x)         
a x cos(x)         
a x tan(x)         
a x sec(x)         
a x cosec(x)   
a x cot(x)         
a x ln(x)           
a x e^(x)       

and finally, just as an example:

f(x)                          f'(x)   

a x sin(bx + c)     
a x cos(bx + c)         
a x tan(bx + c)         
a x sec(bx + c)         
a x cosec(bx + c)     
a x cot(bx + c)         
a x ln(bx + c)               
a x e^(bx + c)         

I'd be eternally greatful if someone could show me how the above are differentiated. I understand it's an extensive list, maybe excessive, so it's perfectly understandable if you don't have the time for me, but I'm just eager to get to grips with this, as to cease resorting to this comunity with similar questions. Thanks so much for your time, I really do appreciate it. David.