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.