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It would be the rate at which temperature changes with time at that specific height, times the rate of change of height with respect to time.That is: This makes perfect intuitive sense: the rates we should consider are the rates at the specified instant.
After we've satisfied our intuition, we'll get to the "dirty work".
We'll learn the step-by-step technique for applying the chain rule to the solution of derivative problems.
Now, let's put this conclusion into more familiar notation.
Let's use the standard letters for functions, f and g.
This rule is usually presented as an algebraic formula that you have to memorize.
There is, though, a physical intuition behind this rule that we'll explore here.
We will look at somewhat more complicated functions in order to show that once we know basic principles, we need not worry.
If you wish to simultaneously follow another text on derivatives in a separate window, click here for Theory and here for Methods Survey.
And if the rate at which temperature drops with height changes with the height you're at (if you're higher the drop rate is faster), T'(h) changes with the height h.
In this case, the question that remains is: where we should evaluate the derivatives?