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This proof has the advantage that it generalizes to several variables.

It relies on the following equivalent definition of differentiability at a point: A function g is differentiable at a if there exists a real number g′(a) and a function ε(h) that tends to zero as h tends to zero, and furthermore , whereas the right-hand side represents the approximation determined by the derivative plus an error term.

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Calling this function η, we have ) tend to zero as h tends to zero, the first two bracketed terms tend to zero as h tends to zero.

Applying the same theorem on products of limits as in the first proof, the third bracketed term also tends zero.

To see this, write the function f(x)/g(x) as the product Because the functions f(g(x)) and x are equal, their derivatives must be equal.

The derivative of x is the constant function with value 1, and the derivative of f(g(x)) is determined by the chain rule.

Because the above expression is equal to the difference The need to define Q at g(a) is analogous to the need to define η at zero.

Carathéodory's alternative definition of the differentiability of a function can be used to give an elegant proof of the chain rule.

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. 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.

It says that if g is a function that is differentiable at a point c (i.e.

the derivative g′(c) exists) and f is a function that is differentiable at g(c), then the composite function It may be possible to apply the chain rule even when there are no formulas for the functions which are being differentiated. If the grade is known, then the rate of ascent can be calculated using trigonometry.

To take the derivative of a composite of more than two functions, notice that the composite of f, g, and h (in that order) is the composite of f with The chain rule can be used to derive some well-known differentiation rules.

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