Ask Uncle Colin: Why does the constant term vanish?

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

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 tinkering1 — 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

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. Why, hello, RM3! What's that? No, nothing to see here. []

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One comment on “Ask Uncle Colin: Why does the constant term vanish?

  • Stephen Cavadino

    This is almost word for word the answer I give my students to the same question!

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