Example 2.9.1. Walking towards a campfire.

You are out in the woods after a long day of mathematics and are walking towards your camp fire on a beautiful still night. The heat from the fire means that the air temperature depends on your position. Let your position at time \(t\) be \(x(t)\text{.}\) The temperature of the air at position \(x\) is \(f(x)\text{.}\) What instantaneous rate of change of temperature do you feel at time \(t\text{?}\)
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  • Because your position at time \(t\) is \(x=x(t)\text{,}\) the temperature you feel at time \(t\) is \(F(t)=f\big(x(t)\big)\text{.}\)
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  • The instantaneous rate of change of temperature that you feel is \(F'(t)\text{.}\) We have a complicated function, \(F(t)\text{,}\) constructed by composing two simpler functions, \(x(t)\) and \(f(x)\text{.}\)
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  • We wish to compute the derivative, \(F'(t) = \diff{}{t} f( x(t) )\text{,}\) of the complicated function \(F(t)\) in terms of the derivatives, \(x'(t)\) and \(f'(x)\text{,}\) of the two simple functions. This is exactly what the chain rule does.
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in-context