Calculus Small Changes, Big Ideas

Section 1.3 The Product Rule and the Chain Rule

• Is the derivative of a product just the product of the derivatives? Can check with units (say price times quantity) and see that it won’t work, and also show that it doesn’t work with \(y=x^2\text{.}\) So we need something new.

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• Visual proof of Product Rule:

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  \begin{align*} \dd(u\cdot v) \amp = \dd{u} \cdot v + u \cdot \dd{v}\\ \amp = u\dd{v} + v\dd{u} \end{align*}

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  \begin{equation*} \dv{x}(f(x)g(x))=f'(x) g(x)+f(x) g'(x) \end{equation*}

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  Can leave the disappearance of \(\dd{f}\dd{g}\) somewhat unresolved, giving a deliberately handwavy explanation of the fact that dividing by \(\dd{x}\) will still leave an infinitesimal quantity in the numerator, so we’ll ignore it. This mystery will get cleaned up in Chapter 5 when we do limits.

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• Check by applying to \(x^{2}\text{:}\)

  \begin{equation*} \dd(x^{2})=x\dd{x} + x\dd{x} = 2x\dd{x} \end{equation*}

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• Extend to three variables:

  \begin{align*} \dd(u\cdot v\cdot w) \amp = \dd((u\cdot v)\cdot w) \\ \amp = \dd(u\cdot v)\cdot w + u\cdot v\cdot \dd{w}\\ \amp = \dd{u}\cdot v\cdot w+u\cdot \dd{v}\cdot w+u\cdot v\cdot \dd{w} \end{align*}

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• Check by applying to \(x^{3}\text{:}\)

  \begin{equation*} \dd(x^{3})=\dd(x\cdot x\cdot x)=3x^{2}\dd{x} \end{equation*}

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• Power Rule:

  \begin{equation*} \dd(x^{n})=nx^{n-1}\dd{x} \end{equation*}

  Only works when \(n\) is a natural number right now, but we can prove it (essentially using induction) by breaking \(x^n\) into \(x^{n-1}\cdot x\) and applying the Product Rule.

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• Justify the Chain Rule by multiplying rates of change and watching the units cancel out. Think of dimensional analysis in high school chemistry.

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• Chain Rule for differentials:

  \begin{equation*} \dd(f(g))=f'(g)\dd{g}=f'(g(x))g'(x)\dd{x} \end{equation*}

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• Chain Rule for derivatives:

  \begin{equation*} \dv{x} f(g(x))=f'(g(x))g'(x) \end{equation*}

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