Related Rates of Change

1. If two variables x and y are connected by the equation y = f(x)

If x changes at the rate of 5 cms ^-1 ⇒  $\frac{dx}{dt}=5$

Two variables, x and y are related by the equation   $y=4x+\frac{3}{x}$ . Given that y increases at a constant rate of 2 units per second, find the rate of change of x when x = 3.

Solution:
$\begin{array}{l}y=4x+\frac{3}{x}=4x+3x^{-1}\\ \frac{dy}{dx}=4-3x^{-2}=4-\frac{3}{x^2}\\ \frac{dy}{dt}=\frac{dy}{dx}\times \frac{dx}{dt}\\ 2=\left(4-\frac{3}{x^2}\right)\times \frac{dx}{dt}\\ \text{when }x=3\\ 2=\left(4-\frac{3}{3^2}\right)\times \frac{dx}{dt}\\ 2=\frac{11}{3}\times \frac{dx}{dt}\\ \frac{dx}{dt}=\frac{6}{11}\text{ unit }{s}^{-1}\end{array}$