How To Solve Derivative Problems

How To Solve Derivative Problems-28
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.So, we know the rate at which the height changes with respect to time, and we know the rate at which temperature changes with respect to height.

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.

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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.

Let's rewrite the chain rule using another notation. According to the rule, if: So, as you can see, the chain rule can be used even when we have the composition of more than two functions.

This rule says that for a composite function: As I said, it is useful for this type of comosite functions to think of an outer function and an inner function. In the previous examples we solved the derivatives in a rigorous manner. Solving derivatives like this you'll rarely make a mistake. In fact, this faster method is how the chain rule is usually applied. We had: If you have just a general doubt about a concept, I'll try to help you.

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?

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.

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.

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