Chapter 2 - Differentiation - 2.3 Exercises - Page 125: 37

$f'(x)=-\frac{4c^2x}{(x-c)^2(x+c)^2}$

Using the quotient rule: $f’(x)=(\frac{u(x)}{v(x)})'=\frac{u'(x)v(x)-v'(x)u(x)}{(v(x))^2}$ $u(x)=(x^2+c^2); u'(x)=2x$ $v(x)=(x^2-c^2); v'(x)=2x$ $f'(x)=\frac{(2x)(x^2-c^2)-(2x)(x^2+c^2)}{(x^2-c^2)^2}$ $=\frac{(2x^3-2c^2x)-(2x^3+2c^2x)}{(x-c)^2(x+c)^2}$ $=-\frac{4c^2x}{(x-c)^2(x+c)^2}$