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Section2.9One 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”).