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Chain Rule, and Higher Derivatives 1

الكلية كلية الهندسة     القسم  الهندسة الميكانيكية     المرحلة 1
أستاذ المادة احمد كاظم حسين الحميري       12/07/2018 16:15:41
It may be possible to apply the chain rule even when there are no formulas for the functions which are being differentiated. This can happen when the derivatives are measured directly. Suppose that a car is driving up a tall mountain. The car s speedometer measures its speed directly. If the grade is known, then the rate of ascent can be calculated using trigonometry. Suppose that the car is ascending at 2.5 km/h. Standard models for the Earth s atmosphere imply that the temperature drops about 6.5 °C per kilometer ascended (called the lapse rate). To find the temperature drop per hour, we apply the chain rule. Let the function g(t) be the altitude of the car at time t, and let the function f(h) be the temperature h kilometers above sea level. f and g are not known exactly: For example, the altitude where the car starts is not known and the temperature on the mountain is not known. However, their derivatives are known: f? is ?6.5 °C/km, and g? is 2.5 km/h. The chain rule says that the derivative of the composite function is the product of the derivative of f and the derivative of g. This is ?6.5 °C/km ? 2.5 km/h = ?16.25 °C/h.

One of the reasons why this computation is possible is because f? is a constant function. This is because the above model is very simple. A more accurate description of how the temperature near the car varies over time would require an accurate model of how the temperature varies at different altitudes. This model may not have a constant derivative. To compute the temperature change in such a model, it would be necessary to know g and not just g?, because without knowing g it is not possible to know where to evaluate f?.

المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .