Question

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

Answer 1

Answer:

A)

$\displaystyle \frac{dy}{dt}=-\frac{33}{8}$

B)

$\displaystyle \frac{dx}{dt}=\frac{3}{2}$

Step-by-step explanation:

x and y are differentiable functions of t, and we are given the equation:

$xy=6$

First, let's differentiate both sides of the equation with respect to t. So:

$\displaystyle \frac{d}{dt}\left[xy\right]=\frac{d}{dt}[6]$

By the Product Rule and rewriting:

$\displaystyle \frac{d}{dt}[x(t)]y+x\frac{d}{dt}[y(t)]=0$

Therefore:

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

A)

We want to find dy/dt when x = 4 and dx/dt = 11.

Using our original equation, find y when x = 4:

$\displaystyle (4)y=6\Rightarrow y=\frac{3}{2}$

Therefore:

$\displaystyle \frac{3}{2}\left(11\right)+(4)\frac{dy}{dt}=0$

Solve for dy/dt:

$\displaystyle \frac{dy}{dt}=-\frac{33}{8}$

B)

We want to find dx/dt when x = 1 and dy/dt = -9.

Again, using our original equation, find y when x = 1:

$(1)y=6\Rightarrow y=6$

Therefore:

$\displaystyle (6)\frac{dx}{dt}+(1)\left(-9)=0$

Solve for dx/dt:

$\displaystyle \frac{dx}{dt}=\frac{3}{2}$