"Just look at these results: the operation of differentiating appears to have had the effect of diminishing the power of x by 1 (for example in the last case reducing x4 to x3), and at the same time multiplying by a number (the same number in fact which originally appeared as the power). Now, when you have once seen this, you might easily conjecture how the others will run. You would expect that differentiating x5 would give 5x4 or differentiating x6 would give 6x5."
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Calculus Made Easy
Calculus Made Easy: being a Very Simplest Introduction to those Beautiful Methods of Reckoning which are Generally Called by the Terrifying Names of the Differential Calculus and the Integral Calculus, by Silvanus Phillips Thompson, was first published in 1910, and is considered a classic and elegant introduction to the subject. A 1998 update by Martin Gardner provides notes for modern readers and provides current versions of many obsolescent mathematical notations or terms. The following quotes
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