How Did I Mess Up? The Answer Should Not Be Negative.

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rayroshi

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This should be an easy one for most people, I guess. In trying to differentiate the given expression, I used the chain rule twice and got an answer that looks almost correct; however, according to the answer sheet provided with the problems, as well as an online derivative calculator, then answer should not be negative. Unfortunately, my answer ends up with a '-' sign in front of it; otherwise, everything else is in accordance with the previously-stated, given correct answers. "Close, but no cigar," as the saying goes.

Where did I go wrong? I have looked at it and looked at it, but I just can't see what I did wrong. Of course, I'm looking at it with the same faulty reasoning as what I had when I did the problem, so I'm in need of a fresh way to look at it.

I have attached a photo of my faulty solution; hope it is clear enough to read. I have never tried to attach a file on this forum, so I don't really know what the required parameters are for such a pic.

Any help would be much appreciated!
 

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Although it is not related to your error, I want to point out that your notation is atrocious:

1701132760599.png

(Yellow is bad, green is good.)

Assuming the point is to differentiate [imath]f(x)=\cot\sqrt[3]{-5x^3-2}[/imath], you are misusing two notations for the derivative:
  • [imath]f'(u)[/imath] does not mean "the derivative of u", and
  • [imath]\frac{dy}{dx}u[/imath] does not mean "the derivative of u".
Rather,
  • [imath]f'(u)[/imath] means "the derivative of function f, evaluated at x = u", and
  • [imath]\frac{dy}{dx}u[/imath] means "the derivative of y with respect to x, times the value of u".
You haven't even defined what f and y are. And using invalid notation means you aren't saying what you mean (to yourself, or to whoever grades your work), and likely that you are not fully understanding what you are trying to express.

What you should be saying on the first line is something like this:

Let [imath]f(x)=\cot\sqrt[3]{-5x^3-2}[/imath]. Then [imath]f'(x)=\frac{d}{dx}\cot\sqrt[3]{-5x^3-2}=\cot'\sqrt[3]{-5x^3-2}\cdot\frac{d}{dx}\sqrt[3]{-5x^3-2}[/imath] and so on.​

In passing, I'll add that I would not have reversed the order of terms; and I probably would have taken [imath]\frac{d}{dx}\sqrt[3]{-5x^3-2}[/imath] off to the side to differentiate it, rather than have to copy the rest several times. These two changes would likely have prevented your actual error!
 
Thank you for your help. Your time and effort are appreciated.

However, I don't see where you addressed my original question: why my answer turned out to be negative, instead of positive. But, with that being said, I saw it this morning. As 'blamocur' said, above, I accidentally dropped the '-' in front of the csc^2, which would've canceled the '-' in front of the '15x^2' yielding the correct, positive answer.
blamocur said:
You dropped the '-' sign in front of [imath]\csc^2...[/imath], which would cancel [imath]-15x^2[/imath]
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Yeah, after a good night's sleep and looking at it again this morning, I saw that, too. Math is so unforgiving! One little misstep, and you're dead. :confused:

Thanks for your time and trouble, though. I really appreciate it!
 
Dr.Peterson said:
Although it is not related to your error, I want to point out that your notation is atrocious:

View attachment 36778

(Yellow is bad, green is good.)

Assuming the point is to differentiate [imath]f(x)=\cot\sqrt[3]{-5x^3-2}[/imath], you are misusing two notations for the derivative:
  • [imath]f'(u)[/imath] does not mean "the derivative of u", and
  • [imath]\frac{dy}{dx}u[/imath] does not mean "the derivative of u".
Rather,
  • [imath]f'(u)[/imath] means "the derivative of function f, evaluated at x = u", and
  • [imath]\frac{dy}{dx}u[/imath] means "the derivative of y with respect to x, times the value of u".
You haven't even defined what f and y are. And using invalid notation means you aren't saying what you mean (to yourself, or to whoever grades your work), and likely that you are not fully understanding what you are trying to express.

What you should be saying on the first line is something like this:

Let [imath]f(x)=\cot\sqrt[3]{-5x^3-2}[/imath]. Then [imath]f'(x)=\frac{d}{dx}\cot\sqrt[3]{-5x^3-2}=\cot'\sqrt[3]{-5x^3-2}\cdot\frac{d}{dx}\sqrt[3]{-5x^3-2}[/imath] and so on.​

In passing, I'll add that I would not have reversed the order of terms; and I probably would have taken [imath]\frac{d}{dx}\sqrt[3]{-5x^3-2}[/imath] off to the side to differentiate it, rather than have to copy the rest several times. These two changes would likely have prevented your actual error!
Click to expand...
 
Well, although I don't understand why your unnecessary, negative comment, I still do appreciate your time and help. I am trying to learn calculus on my own, no class, no teacher, and at an advanced age, so it's a bit tough. All the help that I can get from those like you on forums like this are like gold to me and have been very helpful. I am not a student who is begrudgingly being made to take a hard class in order to satisfy some academic requirement, but rather someone who is just trying to learn for learning's sake.

Regarding the reversal of terms which I did on the third line, I did that strictly because I have often seen that it is done as I tried to follow the step-by-step solution to the problem given by a particular online derivative calculator app. (https://www.derivative-calculator.net/), which I sometimes go to, when trying to follow the logic involved in solving derivative problems. Admittedly, I have no idea why that app sometimes does that, but I thought that maybe it must be in order to present the solutions in a clearer format which would then aid in understanding of the solution. However, from what you said, that must not be so, at least not to you. Are there rules/conventions which dictate the order of writing such complex expressions? The only one that I recall is that order of powers are to be written in descending order, which is, of course, a basic idea.

Thank you.
 
rayroshi said:
Well, although I don't understand why your unnecessary, negative comment,..........
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I am assuming that you felt something negative was pointed at you in response #6.

I did not find anything negative pointed at you in response #6.
 
rayroshi said:
Well, although I don't understand why your unnecessary, negative comment
Click to expand...
You're probably referring to my word "atrocious"; it was not meant negatively, but somewhat humorously (if unwisely). Consider it like telling someone who already knows his English is bad, where his specific errors are. Believe it or not, my intention was to be kind. Thanks for seeing past that line.
rayroshi said:
Admittedly, I have no idea why that app sometimes does that, but I thought that maybe it must be in order to present the solutions in a clearer format which would then aid in understanding of the solution.
Click to expand...
Generally speaking, machines aren't a good guide to good form in writing; it probably has some routine approach to solving certain kinds of problem, and starts out by putting them in the form it knows. It's probably not trained in pedagogy, just correctness.
rayroshi said:
Are there rules/conventions which dictate the order of writing such complex expressions? The only one that I recall is that order of powers are to be written in descending order, which is, of course, a basic idea.
Click to expand...
That sometimes helps you see what you have in a polynomial, by putting like terms together, and important terms first. My focus here is not on "rules" for writing final results, but on how to organize your work so you don't get distracted. A more general idea I often give to students is "clean up the kitchen before you start cooking" -- that is, simplify each side of an equation before you start moving things around. (That doesn't apply here.)

I think my suggestion of taking parts of an equation off to the side to simplify them by themselves is one of the more important heuristics for solving this sort of problem.
 
Okay, thanks for all of that. Electronic communications leave out all of the subtle, but vital, cues such as a smile or body language that are found in a face-to-face conversation, things that we all depend upon, when interpreting others' intentions. But, your time and effort in helping show your kind intent and, as stated, are greatly appreciated. You must spend a good deal of time helping others on this website.

I like your analogy of 'cleaning up the kitchen before you start cooking,' as that says it all. As with most things, I guess, as the saying goes, 'the devil is in the details,' and it will eventually come with time and practice.

Thanks again!
khansaheb said:
I am assuming that you felt something negative was pointed at you in response #6.

I did not find anything negative pointed at you in response #6.
Click to expand...
khansaheb said:
I am assuming that you felt something negative was pointed at you in response #6.

I did not find anything negative pointed at you in response #6.
Click to expand...
Sorry, khansaheb, that wasn't meant to be sent to you. My bad. You answer was great and I thank you for it.
 
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