Chapter 2 - Differentiation - 2.3 Exercises - Page 125: 38

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

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