To find the rate of increase of the $x$-coordinate at point $A(1, -3)$, given the differential equation of the curve and the information about the rate of change of the $y$-coordinate, we need to use the following approach.

Step 1: Understand the given information

We are given:

• The slope of the curve is $\frac{dy}{dx} = \frac{6}{(3x - 2)^3}$.
• At point $A(1, -3)$, the rate of change of the $y$-coordinate, $\frac{dy}{dt}$, is $3$ units per second.

We need to find $\frac{dx}{dt}$ at $A$.

Step 2: Chain rule relation between $\frac{dx}{dt}$ and $\frac{dy}{dt}$

By the chain rule, we know: $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$ This equation connects the rate of change of $y$ with respect to time $t$, to the rate of change of $x$ with respect to time $t$. We can rearrange this to solve for $\frac{dx}{dt}$: $\frac{dx}{dt} = \frac{dy}{dt} \div \frac{dy}{dx}$

Step 3: Substitute known values

At point $A(1, -3)$, we are given $\frac{dy}{dt} = 3$. We need to find $\frac{dy}{dx}$ at $x = 1$. Using the equation for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{6}{(3x - 2)^3}$ Substitute $x = 1$: $\frac{dy}{dx} = \frac{6}{(3(1) - 2)^3} = \frac{6}{(3 - 2)^3} = \frac{6}{1^3} = 6$

Step 4: Calculate $\frac{dx}{dt}$

Now, substitute $\frac{dy}{dt} = 3$ and $\frac{dy}{dx} = 6$ into the chain rule equation: $\frac{dx}{dt} = \frac{3}{6} = \frac{1}{2}$

Thus, the rate of increase of the $x$-coordinate at point $A$ is $\frac{1}{2}$ units per second.

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Would you like further details or clarification?

Here are 5 related questions to consider:

1. How would the rate of change $\frac{dx}{dt}$ vary if $\frac{dy}{dt}$ were different?
2. Can you find the equation of the curve by integrating the slope $\frac{dy}{dx}$?
3. How would you handle the case if $\frac{dy}{dx}$ were more complex, involving trigonometric functions?
4. What happens to $\frac{dx}{dt}$ if the slope $\frac{dy}{dx}$ approaches zero?
5. How can implicit differentiation be applied to problems like this in more complex curves?

Tip: Always apply the chain rule when differentiating a function with respect to time, especially when dealing with rates of change in multiple variables.