This week’s science-related blog has to do with CALCULUS because calculus is awesome and because we actually just got tested on using the Chain Rule on Wednesday. The Physicist over at AskAMathematician.com responded to this question awhile back: “Is there an intuitive proof for the chain rule?” Which, when you think about it, is a really interesting question. We’re taught in calculus that when we’ve got a function “embedded” within another function [e.g., f(g(x))], when we take the derivative of that function, we take the derivative of the “outside function” f’(x) and then multiply it by the derivative of the “inside function” g’(x) to get f’(g(x))g’(x).
But why the heck do we need to do that?
As the Physicist very elegantly points out, it all has to do with slope. When you multiply a function by some amount, you squish it up by that same amount. Using their same example, say you’ve got a function f(x) and it’s “squished” function f(2x). These two functions are the same when x=6 and x=3 respectively, but that’s not the case for their slopes. The squished function 2x will have a steeper one by two. When you take the derivative of this function, then, you have to re-multiply it by two to deal with the squishing.
So how does this work out in general? If you replace the 2x with a more complicated function g(x), you get f(g(x)), and the slope of the whole thing depends, then, on g(x). So when finding the slope (taking the derivative) of f(g(x)), you have to re-multiply it by the slope of g(x) (or the derivative of g(x)) to deal with whatever g(x) is doing to the whole of f(g(x)).
And there’s your chain rule!
Take a look at the actual article. They’ve got pictures and actually have the derivatives and functions written out like I can’t do here.
VERY FREAKING COOL, PEOPLE.