So, what we want is: because the dh "cancel out" in the right side of the equation.Notice that the second factor in the right side is the rate of change of height with respect to time.IT CHANGED MY PERCEPTION TOWARD CALCULUS, AND BELIEVE ME WHEN I SAY THAT CALCULUS HAS TURNED TO BE MY CHEAPEST UNIT.Tags: Math Homework Problem SolverPenghargaan AssignmentIn An Essay Plays Are UnderlinedInterview Questions Problem SolvingNsf Dissertation GeographyEssay On EvolutionEducation In 2050 Essay
The result in our concrete example coincides with this differentiation rule: the rate of change of temperature with respect to time equals the rate of temperature vs. In our example we have temperature as a function of both time and height.
We know the derivative of temperature with respect to height, and we want to know its derivative with respect to time.
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.
In this page we'll first learn the intuition for the chain rule.
This intuition is almost never presented in any textbook or calculus course.
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That will be simply the product of the rates: if height increases 1 km for each hour, and temperature drops 5 degrees for each km, height changes 5 degrees for each hour. We know the derivative equals the rate of change of a function, so, what we concluded in this example is that if we consider the temperature as a function of time, T(t), its derivative with respect to time equals: In the previous example the derivatives where constants.
We set a fixed velocity and a fixed rate of change of temperature with resect to height. If, for example, the speed of the car driving up the mountain changes with time, h'(t) changes with time.