I have the following function: $$f(x)=\frac{[2a x^2-(a-1)][(a-1)x^2+2]}{(a+1)^2x^2}$$ where $a>1$ is a constant.
I want to prove that the Taylor expansion of $f$ in the neighbourhood of $x=1$ is $$f(x)\approx 1+4\frac{a-1}{a+1}(x-1).$$ I know how to this by calculating the first derivative of $f$ at $x=1$ and substituting its value into the Taylor series. However, I would like to know if there is another way to solve the problem without using the derivatives. I tried expanding the numerator and denominator of $f$ up to the first order in $(x-1)$, using $$x^2=[1+(x-1)]^2\approx1+2(x-1)$$ and substituted this into $f$, but then I got a wrong result. Can anyone help with this please?