Chapter 2 - Differentiation - 2.3 Exercises - Page 125: 42

$f'(x)=\frac{-3\sin(x)+x\cos(x)}{x^4}$ .

Using the quotient rule: $f’(x)=(\frac{u(x)}{v(x)})'=\frac{u'(x)v(x)-v'(x)u(x)}{(v(x))^2}$ $u(x)=\sin(x); u'(x)=\cos(x)$ $v(x)=x^3; v'(x)=3x^2$ $f'(x)=\frac{(\cos(x))(x^3)-(3x^2)(sin(x))}{(x^3)^2}$ $=\frac{(x^2)(x\cos(x)-3\sin(x))}{x^6}$ $=\frac{-3\sin(x)+x\cos(x)}{x^4}$