Product rule over qoutient rule

A

abdurrehman

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Oct 29, 2019
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my question is can we always prevent qoutient rule by using product rule ?
equation like
✓ (x-1)/x
✓ e/e^x-1
 
1) Short answer. Yes.
2) Better answer. Why? Do not EVER limit your tools or abilities. Get fluent at both. With time and experience you will see that each has its space for improving your life.

Note: You may mean "avoid" rather than "prevent".
 
abdurrehman said:
my question is can we always prevent qoutient rule by using product rule ?
equation like
✓ (x-1)/x
✓ e/e^x-1
Click to expand...
In effect, using the product rule where the quotient rule applies amounts to re-deriving the quotient rule. (Therefore, it can always be done.) Sometimes that takes less work, sometimes more. I go to the product rule most often if I haven't been doing much calculus lately and am not positive that I will get the signs right! That's usually a bigger issue than the amount of work.

In your first example, I'd distribute first, and use neither rule! I suspect you haven't written what you intend in the second example; did you mean (e/e^x) - 1 as written, or e/(e^x - 1), or e/e^(x-1)?
 
Yes, since A/B can always be written as A * (1/B) or A * B-1 which is a product.
The question is why would you want to do this?
 
JeffM said:
To avoid sign errors. Plus it is one less rule to memorize.
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I understand what you are saying and agree with it up to a point. My concern is that students have difficulty using the quotient rule even when the rule is in front on them. I think that students should be able to do the algebra! When students learn how to deal with certain algebra equations/expressions when they couldn't before this benefits them.
 
You have to do the same algebra either way. I just find that the product rule is memorable whereas the quotient rule is not and so leads to unnecessary error.
 
JeffM said:
You have to do the same algebra either way. I just find that the product rule is memorable whereas the quotient rule is not and so leads to unnecessary error.
Click to expand...
I agree entirely. Especially if there are a lot of terms to keep track of.

-Dan
 
thanks everyone for clarifying but i am stuck on this particular problem which i attached.
i get different answer...
 

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JeffM said:
You have to do the same algebra either way. I just find that the product rule is memorable whereas the quotient rule is not and so leads to unnecessary error.
Click to expand...
Yes, you have to do the algebra in both cases. Exactly what I am saying is that students have more trouble simplifying the results from the quotient rule than the product rule. I am not sure why this is the case but I have seen it while working about 10,000 hours in tutoring labs while in college.
 
Jomo said:
Yes, you have to do the algebra in both cases. Exactly what I am saying is that students have more trouble simplifying the results from the quotient rule than the product rule. I am not sure why this is the case but I have seen it while working about 10,000 hours in tutoring labs while in college.
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That is a really odd but fascinating psychological observation.

The human mind works in very strange ways. Computers must view us as systematically insane.
 
Here's the way I remember, and encourage my students to remember, the quotient rule:

If \(\displaystyle y = \frac{u}{v}\) then \(\displaystyle y' = \frac{vu' -uv'}{v^2} \).

Note the "superscripts" on the v's ... they increase as you write the RHS out (no superscript on first v, 1( actually ' ) superscript on second v, 2 superscript on last v). And you've got to remember a minus sign on the top.

Always worked for me.
 
My trouble is that, though I may memorize the rule in one particular way (my current preference is [MATH]y' = \frac{u'v - v'u}{v^2}[/MATH], where I differentiate the top first, then the bottom, for which reason I prefer the product rule the same way, [MATH]y' = u'v + v'u[/MATH] -- when I am tutoring, a student may have learned a different version of the formula, and I have to work with what they have learned. As a result, my memory keeps getting reset!

But I agree both that the formula is tricky to get right and that it can be harder to work with the resulting form ... sometimes.
 
abdurrehman said:
thanks everyone for clarifying but i am stuck on this particular problem which i attached.
i get different answer...
Click to expand...
You have a basic error in your very first step. What is the derivative of x^-1? It is not (-x)^2 but -(x^-2). On the next line, you evidently treat it as x^-2, but you don't have the negative.

You need to be more careful about parentheses, both there and elsewhere!
 
Harry_the_cat said:
Here's the way I remember, and encourage my students to remember, the quotient rule:

If \(\displaystyle y = \frac{u}{v}\) then \(\displaystyle y' = \frac{vu' -uv'}{v^2} \).

Note the "superscripts" on the v's ... they increase as you write the RHS out (no superscript on first v, 1( actually ' ) superscript on second v, 2 superscript on last v). And you've got to remember a minus sign on the top.

Always worked for me.
Click to expand...
I tell my students that the rule for the quotient rule is the same as the product rule except the sign in the middle is negative and you divide by the denominator squared. I am always amused at how many calculus books write the product rule in such a way that it does NOT match the quotient rule as I just described.
 
Dr.Peterson said:
You have a basic error in your very first step. What is the derivative of x^-1? It is not (-x)^2 but -(x^-2). On the next line, you evidently treat it as x^-2, but you don't have the negative.

You need to be more careful about parentheses, both there and elsewhere!
Click to expand...


Thank you very much for clarifying...... now i got my mistake...
 
yes it
Dr.Peterson said:
My trouble is that, though I may memorize the rule in one particular way (my current preference is [MATH]y' = \frac{u'v - v'u}{v^2}[/MATH], where I differentiate the top first, then the bottom, for which reason I prefer the product rule the same way, [MATH]y' = u'v + v'u[/MATH] -- when I am tutoring, a student may have learned a different version of the formula, and I have to work with what they have learned. As a result, my memory keeps getting reset!

But I agree both that the formula is tricky to get right and that it can be harder to work with the resulting form ... sometimes.
Click to expand...
yes it is bit tricky .. things are easy if you prefer to use relative rule on the problem if you manuplate the problem in your own way than things can difficult and tricky like i was trying to use product rule where i can easily apply quotient rule
 
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