# Derivative problem from old book, is the answer a typo, if not, why not?

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15 minutes ago, Zero Cool said:

This problem is from the book Calculus Made Easy by Silvanuws P. Thompson, copyright 1946 (3rd Ed.)

Silvanus' answer is correct and follows the standard rule for differentiation of a power.

if  $u = {t^p}$

then

$\frac{{du}}{{dt}} = p{t^{\left( {p - 1} \right)}}$

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Yeah. Everything OK IMO.

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1 hour ago, studiot said:

Silvanus' answer is correct and follows the standard rule for differentiation of a power.

if  u=tp

then

dudt=pt(p1)

Ok, so p = 2, and not 8, but since they are multiplied, that doesn't make sense to me. I have been out of school for a long time. Now (book is 75years old), would we use some parenthesis or something to show the 2 and the 4 are separate entities?

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p=2.4.  The dot is a decimal point not the multiplication operator. so u=t^(2.4) and du/dt=2.4*t^(2.4-1=1.4).
hope I didn't make things worse, I have the same book and found the notation difficult too.

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10 hours ago, Zero Cool said:

Ok, so p = 2, and not 8, but since they are multiplied, that doesn't make sense to me. I have been out of school for a long time. Now (book is 75years old), would we use some parenthesis or something to show the 2 and the 4 are separate entities?

No parenthesis was used so it should be taken to mean p = 2.4.

Strictly to make the difference and indicated multiplication your could (perhaps should put the 2 and the 4 each in their own brackets, but I have never seen this done in that way.

I am a little surprised at your continuing difficulty since you yourself identified that the bold dot is used by Silvanus to signify both the decimal point and the operation of multiplication.

13 hours ago, Zero Cool said:

The author uses a dot for showing both multiplication and as a divider between the ones column and the tenths column (see pics)

The original book by Silvanus was published in 1910 and at that time there was a fashion for printers to use the bold dot in the middle of the line to avoid confusion a small dot being lost at the bottom of the line.
This bold dot symbol is still available today and I have shown it in my revision examples of the use of symbols for powers and roots below.
Note I have used the modern star in the middle to represent multiplication.
The star was introduced to avoid the ambiguity of using one symbol (the dot) for two different things.

${t^{2 \bullet 4}} = {t^{2.4}} = \sqrt[{10}]{{{t^{24}}}}$

${t^2}*{t^4} = {t^{\left( {2 + 4} \right)}} = {t^6}$

${t^{\left( {2*4} \right)}} = {t^{2 \bullet 4}} = {\left( {{t^2}} \right)^4} = {t^8}$

I hope these help

Edited by studiot
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• 2 weeks later...

You, yourself, said, in your first post, "The author uses a dot for multipllication and as a divider between the ones column and the tenths column" (I would have said "as a decimal point").   In this problem the author is using it as a decimal point but you are interpreting it as a multiplication.

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Notice that the author  does NOT use the same "dot" for multiplication and a decimal point.  Multiplication is indicated by a centered dot ($2\cdot 4$) while the decimal point is on the baseline (2.4).

Edited by Country Boy

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