A student differentiates $$y=\frac{(x^2+1)^2}{x-1},\qquad x\ne1,$$ as follows: \begin {aligned} \fra

Question

A student differentiates $$y=\frac{(x^2+1)^2}{x-1},\qquad x\ne1,$$ as follows: \begin {aligned} \frac{dy}{dx}&=\frac{4x(x^2+1)}1\ &=4x(x^2+1).\end {aligned} Explain the student’s error. Find the correct derivative in factorised form. Find the actual gradient at $x=2$, and compare it with the student’s answer.

Accepted Answer

(b) $y'=\frac{(x^2+1)(3x^2-4x-1)}{(x-1)^2}$; (c) Actual gradient $15$, student’s value $40$.

Differentiation

Differentiation