Question

Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt.

xy = 2

a. Find dy/dt, when x = 4, given that dx/dt = 13.
b. Find dx/dt, when x = 1, given that dy/dt = -9.

Answer 1

Answer:

a. $\frac{dy}{dt} = -\frac{13}{8}$

b. $\frac{dx}{dt} = \frac{9}{2}$

Step-by-step explanation:

To solve this question, we apply implicit differentiation.

xy = 2

Applying the implicit differentiation:

$y\frac{dx}{dt} + x\frac{dy}{dt} = \frac{d}{dt}(2)$

$y\frac{dx}{dt} + x\frac{dy}{dt} = 0$

a. Find dy/dt, when x = 4, given that dx/dt = 13.

x = 4

So

$xy = 2$

$4y = 2$

$y = \frac{2}{4} = \frac{1}{2}$

Then

$y\frac{dx}{dt} + x\frac{dy}{dt} = 0$

$\frac{1}{2}(13) + 4\frac{dy}{dt} = 0$

$4\frac{dy}{dt} = -\frac{13}{2}$

$\frac{dy}{dt} = -\frac{13}{8}$

b. Find dx/dt, when x = 1, given that dy/dt = -9.

x = 1

So

$xy = 2$

$y = 2$

Then

$y\frac{dx}{dt} + x\frac{dy}{dt} = 0$

$2\frac{dx}{dt} - 9 = 0$

$2\frac{dx}{dt} = 9$

$\frac{dx}{dt} = \frac{9}{2}$