Dear Uncle Colin,

When I differentiate $y=2x^2 + 7x + 2$ and apply the $nx^{n-1}$ rule, why do I only apply it to the $2x^2$ and the $7x$ but not the 2?

-- Nervous Over Rules, Mathematically A Liability

Hi, NORMAL, and thanks for your message!

There are several ways to answer that, but I’ll limit myself to two.

The first is to say “actually, you do apply the rule to the 2.” The constant term in your polynomial isn’t just the number 2, but really $2x^0$. When you apply the stated rule to that, you get $0\times 2x^{-1}$ – and ignoring a local difficulty at $x=0$, which can be cleared up with a bit of deep-level tinkering ((Why, hello, RM3! What’s that? No, nothing to see here.)) – that’s always 0. That means we can ignore constant terms when differentiating.

Now, I don’t much like the rule – only partly because of that local difficulty. Instead, I like to think about what the constant term does to the gradient of the curve. If you move the entire curve up or down two units, the gradient at any given $x$-value doesn’t change - so the constant term makes no contribution to the gradient, and can be ignored when differentiating.

Hope that helps!

-- Uncle Colin