A chain of functions starts with y = g(x)   Then it finds z = f(y).  So z = f(g(x))
Very many functions are built this way, g inside f .   So we need their slopes.

The Chain Rule says :  MULTIPLY THE SLOPES  of  f and g.

Find dy/dx for g(x).   Then find dz/dy for f(y). 
Since dz/dy is found in terms of y, substitute g(x) in place of y !!!
The way to remember the slope of the chain is dz/dx = dz/dy times dy/dx. 
Remove y to get a  function of x !   The slope of z = sin (3x) is 3 cos (3x).

Professor Strang’s Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.