Is this answer wrong? Shouldn't it be $+8x$ in numerator? (it's in my textbook)

Question

I did it with log and quotient rule and got $+8x$ in numerator both times. Am I making a mistake or is the answer is wrong?

Answer 1

On the third to last line the book calculates $(x^3 + 4x) - (x^3 - 4x) = -8x$.

Well.... obviously that is a careless error. (One that we've all made and will all make again sometime).

So you are right and the book is wrong.

This may be overkill but $x > 2$ then book's derivative is negative and ours is positive. So for $2 < x < x'$ we can check to see if $y$ increases or decreases.

I claim for positive $b > a > c$ that $\frac {b-c}{b+c} > \frac {a-c}{a+c} \iff (b-c)(a+c) > (b+c)(a-c) \iff bc - ac > ac - bc $ which is clearly true as $bc - ac > 0 > ac - bc$.

So $\sqrt{\frac {x'^2 -4}{x'^2 + 4}} > \sqrt{\frac {x^2 -4}{x^2 +4}}$ and $y$ is increasing which confirms our suspicions and refutes the book's result.