Chapter 3 - Differentiation - 3.8 Implicit Differentiation - Exercises - Page 154: 59

See the answer below.

Implicitly differentiating $xy=1$ with respect to $t$, we have $\frac{dx}{dt}\times y+ x\frac{dy}{dt}=0$ $\implies x \frac{dy}{dt}=-y\frac{dx}{dt}$ $\implies \frac{dy}{dt}=-\frac{y}{x}\frac{dx}{dt}$