Solution assignment 24 Product and Quotient rule

Differentiate:

$y=\displaystyle\frac{x+4}{x^2-16}$ , $x\neq4$ and $x\neq{-4}$

It is faster to simplify this function first. We can write by factorizing the denominator:

$y=\displaystyle\frac{x+4}{(x+4)(x-4)}=\displaystyle\frac{1}{x-4}$

We know:

$f'(x)=0$ and $g'(x)=1$

and thus the quotient rule yields:

$y'=\displaystyle\frac{(x-4)\cdot0-1.1}{(x-4)^2}=-\displaystyle\frac{1}{(x-4)^2}$

Note that both the original function and the simplified function are discontinuous for $x=4$ and $x=-4$ and cannot be differentiated in these points.