Before the midterm

Misforming the quotient rule

The habit: Low d-high minus high d-low, over low squared: the order in the numerator sets the sign, and the denominator is squared.

17 practice questions are built to catch it, in 6 topics.

What it looks like

A question from differentiation rules, with the answer this slip gives beside the right one.

Introductory2 marks · 3 min

A student differentiates $f(x) = \dfrac{5x}{x^{2} + 1}$ as shown below. Which line contains the first error?

\begin{aligned} \text{Line 1:}\quad f'(x) &= \frac{5\,(x^{2} + 1) + 5x\cdot 2x}{(x^{2} + 1)^{2}} \\[6pt] \text{Line 2:}\quad &= \frac{5x^{2} + 5 + 10x^{2}}{(x^{2} + 1)^{2}} \\[6pt] \text{Line 3:}\quad &= \frac{5\,(3x^{2} + 1)}{(x^{2} + 1)^{2}} \end{aligned}

With the slip: There is no error: the quotient rule adds u′v and uv′, and lines 2 and 3 are correct.
Worked through: Line 1: the quotient rule subtracts, so the numerator should be 5(x² + 1) − 5x · 2x.

(u′v − uv′)/v² subtracts. Compare each line’s numerator with that, starting from the first.

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.