The Chain Rule The Chain Rule In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function f ∘ g in terms of the derivatives of f and g. In integration, the counterpart to the chain rule is the substitution rule. The chain rule seems to have first been used by Leibniz. He used it to calculate the derivative of as the composite of the square root function and the function . He first mentioned it in a memoir with various mistakes in it. The common notation of chain rule is due to Leibniz. L'Hôpital uses the chain rule implicitly in his Analyse des infiniment petits but also does not state it explicitly. The chain rule does not appear in any of Leonhard Euler's analysis books, even though they were written over a hundred years after Leibniz's discovery. In the statement of the chain rule, f and g play slightly different roles because f′ is evaluated at g(t) whereas g′ is evaluated at t. This is necessary to make the units work out correctly. For example, suppose that we want to compute the rate of change in atmospheric pressure ten seconds after the skydiver jumps. This is (f ∘ g)′(10) and has units of Pascals per second.
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The factor g′(10) in the chain rule is the velocity of the skydiver ten seconds after his jump, and it is expressed in meters per second. f′(g(10)) is the change in pressure with respect to height at the height g(10) and is expressed in Pascals per meter. The product of f′(g(10)) and g′(10) therefore has the correct units of Pascals per second. It is not possible to evaluate f anywhere else. For instance, because the 10 in the problem represents ten seconds, the expression f′(10) represents the change in pressure at a height of ten seconds, which is nonsense. Similarly, because g′(10) = −98 meters per second, the expression f′(g′(10)) represents the change in pressure at a height of −98 meters per second, which is also nonsense. However, g(10) is 3020 meters above sea level, the height of the skydiver ten seconds after his jump. This has the correct units for an input to f. The chain rule in the absence of formulas 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 (see 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.
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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′.
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