# Connected rates of change calculus.

I'm struggling with connected rates of change $\frac{dy}{dt} =\frac{dy}{dx}\times \frac{dx}{dt}$ I'll give you an example of the type of questions I'm struggling with and I'll explain what I don't understand about the question.

Question: Variables x and y are connected by the equation

$y=x+\sqrt{x-5} $.

Given that x increases at a rate of 0.1 units per second. Find the rate of change of y when x = 9.

When I initially read this question, it feels like the question is asking me to do $y=(9+ \sqrt{9-5} )\times 0.1 = 1.1$ which is incorrect, so my first question is what is the question asking and how does it have anything to do with a curve/tangent?

My second question is in my book, it tells me to solve the problem like this

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

$\frac{dx}{dt} =0.1$

$1+\tfrac{1}{2\sqrt{9-5}}\times 0.1 =0.125$

why is $\tfrac{dx}{dt} = 0.1 $ because my understanding of differentiation is that for instance if you have $y=x^3 $ and then you find the derivative of y with respect to x what it's asking is the gradient of y at any given value of x so shouldn't that mean that $\tfrac{dx}{dt}$ is asking for the value of x at any given value of t rather than an actual number if that makes sense.

If you don't understand exactly what I mean I'll try to explain further.

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