Question

a point is moving on the graph of xy = 12. when the point is at (3,4), its x - coordinate is increasing by 3 units per second. how fast is the y - coordinate changing at that moment? the y - coordinate is at units per second. (simplify your answer.)

Explanation:

Step1: Differentiate the equation

Differentiate $xy = 12$ with respect to time $t$ using the product - rule $(uv)^\prime=u^\prime v + uv^\prime$. Here $u = x$ and $v = y$. So we get $x\frac{dy}{dt}+y\frac{dx}{dt}=0$.

Step2: Substitute the given values

We know that $x = 3$, $y = 4$, and $\frac{dx}{dt}=3$. Substitute these values into the equation $x\frac{dy}{dt}+y\frac{dx}{dt}=0$. We have $3\frac{dy}{dt}+4\times3 = 0$.

Step3: Solve for $\frac{dy}{dt}$

First, simplify the equation $3\frac{dy}{dt}+12 = 0$. Then subtract 12 from both sides: $3\frac{dy}{dt}=-12$. Divide both sides by 3 to get $\frac{dy}{dt}=- 4$.

Answer:

decreasing; 4