The Mathematical Pirate and The Quotient Rule

"Arr, that be a scurvy-lookin' expression!" said the Mathematical Pirate. "A quartic on the top and a quadratic on the bottom. That Ninja would probably try to factorise and do it all elegant-like."

"Is that not the point?"

"When you got something as 'orrible as that, it's like puttin' lipstick on a shark. There ain't all that much point, and it's just gonna annoy the shark."

The student looked both ways. It was not a metaphor he was familiar with. "Right you are, then, captain. So we've got to find the gradient of $f(x) = \frac{x^4 + x^3 - 13x^2 + 26x - 17}{x^2 - 3x + 3}$ when $x=1$. That's looking all quotient-ruley to me."

"Arr," nodded the Mathematical Pirate.

"I mean, $u$ and $v$ on their own look ok, and $u'$ and $v'$1 are even nicer, but their products..."

"Agreed!" agreed the Mathematical Pirate. "Work them out anyway!"

"Ok. $u = x^4 + x^3 - 13x^2 + 26x - 17,$, so $u' = 4x^3 + 3x^2 - 26x + 26$; $v = x^2 - 3x + 3$, so $v' = 2x - 3$."

"Now figure those out when $x=1$."

The student's eyes opened wide. "You mean..."

"I mean. You don't have to do all that algebra to find the gradient - you can just put a value in and use the quotient rule on the result. Works for the other rules, too."

The student looked hurt, mildly offended, and then overjoyed.

"So, $u = -2$, $u' = 7$, $v = 1$ and $v' = -1$. That's already a lot nicer. $\frac{vu' - uv'}{v^2} = \frac{7 - 2}{1} = 5 $. Is that it?!"

"Ahar," said the Mathematical Pirate. "Don't tell the Ninja."

* Edited 2020-01-16 because I'd said "Ninja" instead of "Pirate" and now I fear for my life. Thanks a bunch, Adam.


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. The Mathematical Pirate uses Newton's notation, just to annoy the Ninja []


One comment on “The Mathematical Pirate and The Quotient Rule

  • Stephen Cavadino

    I do like this shortcut. Falls out nicely when you have a firm grasp of algebra.

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