Helping Ada Lovelace with her Homework: Classroom Exercises from a Victorian Calculus Course – Solution to Problem 5

To set up De Morgan's approach to solving Problem 5, let \(u\) be the product of \(n\) functions, \(f_1(x),f_2(x), \ldots ,f_n(x)\), where \(f_1(x)=f_2(x)=\ldots = f_n(x)=x\).  In other words, let \(u=x^n\).  Then using De Morgan’s version of Peacock’s Theorem 2:

\[\Large{ \begin{array}{lcl} \frac{du}{dx} &=& \frac{u}{f_1(x)}\cdot \frac{df_1}{dx} + \frac{u}{f_2(x)}\cdot\frac{df_2}{dx} + \frac{u}{f_3(x)}\cdot\frac{df_3}{dx} + \ldots + \frac{u}{f_n(x)}\cdot\frac{df_n}{dx} \\ \\  &=& \frac{x^n}{x }\cdot \frac{dx}{dx} + \frac{x^n}{x} \cdot \frac{dx}{dx} + \frac{x^n}{x} \cdot \frac{dx}{dx} + \ldots + \frac{x^n}{x} \cdot \frac{dx}{dx}\\ \\ &=& x^{n-1}+x^{n-1}+x^{n-1}+\ldots+x^{n-1}  \,\,\,\, (n \mbox{ times}) \\ \\ &=&nx^{n-1}. \end{array} }\]

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