why is the derivative of a composite function the product of two derivatives?

Question

why is the derivative of a composite function the product of two derivatives? ? Don't need much detail, just explain in simplest form. Thanks.

Answer 1

Disclaimer: This is a nice way to think about it, but mathematically is pretty sketchy and kind of hand-wavy.  Seems like it's what you're looking for though.  If not let me know.

 

If you're looking at a composite function like f(g(x)) and you want to take the derivative, your goal is to find df/dx.  The problem is that g is in the way of that.  So we must first find the derivative of f in terms of g and then multiply that by the derivative of g in terms of x.  Because when we do that we end up with

 

(df/dg)(dg/dx) = df/dx  (basically the dg's cancel...  which is the not so mathy, hand-wavy part because g is a function)

 

But it should help you visualize whats happening.

 

For example if you have f(g(h(x))) and you want the derivative of f, you can do this:

 

(df/dg)(dg/dh)(dh/dx) = df/dx    

So you need to find the three derivatives as it goes deeper and deeper into the function and multiply them all.