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Section 2.9 One More Tool – the Chain Rule

We have built up most of the tools that we need to express derivatives of complicated functions in terms of derivatives of simpler known functions. We started by learning how to evaluate

  • derivatives of sums, products and quotients
  • derivatives of constants and monomials

These tools allow us to compute derivatives of polynomials and rational functions. In the previous sections, we added exponential and trigonometric functions to our list. The final tool we add is called the chain rule. It tells us how to take the derivative of a composition of two functions. That is if we know \(f(x)\) and \(g(x)\) and their derivatives, then the chain rule tells us the derivative of \(f\big(g(x)\big)\text{.}\)

Before we get to the statement of the rule, let us look at an example showing how such a composition might arise (in the “real-world”).

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 \(t\text{?}\)

  • Because your position at time \(t\) is \(x=x(t)\text{,}\) the temperature you feel at time \(t\) is \(F(t)=f\big(x(t)\big)\text{.}\)
  • 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 \(f(x)\text{.}\)
  • 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.