# How To Solve Derivative Problems 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.

## Comments How To Solve Derivative Problems

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Solved Problems for Derivative. Here we will show how derivatives are calculated. 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.…

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The first derivative is used to minimize the surface area of a pyramid with a square base. A detailed solution to the problem is presented. Solve Tangent Lines Problems in Calculus. Tangent lines problems and their solutions are presented. Solve Rate of Change Problems in Calculus. Calculus Rate of change problems and their solutions are presented.…

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This series shows how to solve several types of Calculus limit problems. Special cases of limits are solved and the related graphs are described. Solving Calculus limit and derivative problems are made understandable in this guide. See examples of how to find the derivative using derivative rules.…

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Ht takes the shape as a cartesian graph of a parabola with vertex as a maximum. The positive root will be the time when the ball reaches the ground. You can use the first derivative of h, equate this derivative to zero, solve this for t, and that is the time when h is the maximum.…

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In calculus, the way you solve a derivative problem depends on what form the problem takes. Common problem types include the chain rule; optimization; position, velocity, and acceleration; and related rates. Here are a few things to remember when solving each type of problem Chain Rule problems Use the chain rule when the argument of…

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