as f(x)=1/(x^2-1)f(x)=1x2−1
f(x+h)=1/((x+h)^2-1)f(x+h)=1(x+h)2−1
Hence, f(x+h)-f(x)=1/((x+h)^2-1)-1/(x^2-1)f(x+h)−f(x)=1(x+h)2−1−1x2−1
= ((x^2-1)-((x+h)^2-1))/((x^2-1)((x+h)^2-1))(x2−1)−((x+h)2−1)(x2−1)((x+h)2−1)
= ((x^2-1)-(x^2+2hx+h^2-1))/((x^2-1)((x+h)^2-1))(x2−1)−(x2+2hx+h2−1)(x2−1)((x+h)2−1)
= ((x^2-1-x^2-2hx-h^2+1))/((x^2-1)((x+h)^2-1))(x2−1−x2−2hx−h2+1)(x2−1)((x+h)2−1)
= ((-2hx-h^2))/((x^2-1)((x+h)^2-1))(−2hx−h2)(x2−1)((x+h)2−1) and
(f(x+h)-f(x))/h=((-2x-h))/((x^2-1)((x+h)^2-1))f(x+h)−f(x)h=(−2x−h)(x2−1)((x+h)2−1)
Now (df)/(dx)=Lt_(h->0)(f(x+h)-f(x))/hdfdx=Lth→0f(x+h)−f(x)h
= Lt_(h->0)((-2x-h))/((x^2-1)((x+h)^2-1))Lth→0(−2x−h)(x2−1)((x+h)2−1)
= (-2x)/((x^2-1)(x^2-1))−2x(x2−1)(x2−1)
= (-2x)/(x^2-1)^2−2x(x2−1)2