Question

For each of the following equations connecting x and y, if the rate of change of x is 2units, find the rate of change of y at the given instant

a) y= 3/(2x-3)³, x=2

b) y= x³ + 2 , y=10

Answer 1

given, the rate of change of x is 2units i.e.,$\frac{dx}{dt}=2$ .$a)\\ y=\frac{3}{(2x-3)^3}, at\space x=2\\ \frac{dy}{dt}=3.(-3)(2x-3)^{-2}.(2\frac{dx}{dt})\\ \frac{dy}{dt}=(-9)(2\times2-3)^{-2}.(2\times2)\space(\text{Since, x=2 and rate of change of x is 2 units given.})\\ \frac{dy}{dt}=-36\\ \text{Therefore, the rate of change of y at the given instant is -36.}\\ b)\\ y= x^3 + 2 , y=10 \\ \frac{dy}{dt}=3x^2.\frac{dx}{dt}\\ \frac{dy}{dt}=3(10)^2\times2\\ \frac{dy}{dt}=600\\ \text{Therefore, the rate of change of y at the given instant is 600.}\\$