Connected rates of change calculus.

I understand the confusion you're facing. Let's break it down and I’ll clarify the concepts involved.

The first question is asking you to find the rate of change of y with respect to time (dy/dt) when x is equal to 9, given that x is increasing at a rate of 0.1 units per second (dx/dt = 0.1). In other words, as x changes at a certain rate (dx/dt), how does y change in response to that rate (dy/dt) at a specific point (x = 9)?

In terms of the relationship between dy/dt and dx/dt, the key concept in this problem is related rates, which involves finding the rate of change of one variable with respect to another variable. In this case, you're given dy/dx (rate of change of y with respect to x) and dx/dt (rate of change of x with respect to t). The relationship between these rates is given by the chain rule:

$\frac{dy}{dt} =\frac{dy}{dx}\times \frac{dx}{dt}$

This formula allows you to find how y changes with respect to time when you know the rates of change of x and y with respect to each other.
Now, how to find dy/dx?

To calculate dy/dx, you need to find the derivative of y with respect to x. Given that:

$y=x+\sqrt{x-5}  $

You correctly calculated dy/dx as:

$\frac{dy}{dx} = 1 +\frac{1}{2\sqrt{x-5} } $

Now, you have the rate of increase in y with respect to x. The question wants that rate to be calculated at a specific value of x. So, you need to evaluate dy/dx at x = 9, so plug x = 9 into the expression, you’ll have:

$\frac{dy}{dx} = 1 +\frac{1}{2\sqrt{9-5} } $

$\frac{dy}{dx} =\frac{5}{4} $


Now, how to find dy/dt?

Finally, use the chain rule formula to find dy/dt:

$\frac{dy}{dt} =\frac{dy}{dx}\times \frac{dx}{dt}$

$\frac{dy}{dt} =\frac{5}{4} \times 0.1$

$\frac{dy}{dt} = 0.125$


So, when x = 9 and dx/dt = 0.1, the rate of change of y with respect to time (dy/dt) is 0.125 units per second.

In this context, dx/dt is not the value of x at any given value of t; rather, it represents the rate at which x is changing with respect to time t. In the given problem, dx/dt = 0.1 means that x is increasing at a rate of 0.1 units per second.


Now, I explain the concept a bit more, for better understanding:

Conceptual explanation of rate of change, tangent line, and curve

The concept of related rates is closely connected to curves and tangents in calculus. When you are dealing with related rates problems, you are essentially examining how two variables that are connected by an equation change with respect to each other.

In the given problem, you have the equation that connects the variables x and y:

$y=x+\sqrt{x-5}  $

This equation represents a curve in the Cartesian coordinate plane. Each point on this curve corresponds to a particular value of x and y that satisfy the equation. As x changes, the corresponding y values on the curve will also change, and this change is determined by the given equation.

Now, let's focus on a specific point on this curve where x = 9. At x = 9, the equation gives you a corresponding y value. As x changes very slightly, the y value will change too, and this change can be represented by a tangent line at the point (9, y) on the curve.

The rate of change of x with respect to time (dx/dt) tells you how fast the x-coordinate is changing at that specific point (9, y). It gives you the slope of the tangent line at that point with respect to the x-axis.
Similarly, the rate of change of y with respect to time (dy/dt) tells you how fast the y-coordinate is changing at the same point (9, y). It gives you the slope of the tangent line at that point with respect to the y-axis.

So, the related rates problem is essentially asking you to find the rate of change of y with respect to time (dy/dt) when x = 9, considering the tangent line to the curve at that particular point. The value of dy/dt in this context represents the instantaneous rate of change of y as x changes at a constant rate (0.1 units per second). This is why we use the chain rule and related rates concepts to find the relationship between dy/dt and dx/dt in this problem.