It is a difference in how the function is presented before differentiating (or how the functions are presented). Y = -3/5x+7/5 gives y explicitly as a function of x. 3x+5y=7 gives exactly the same relationship between x and y, but the function is implicit (hidden) in the equation. To make the function explicit, we solve for x In x^2+y^2=25, y is not a function of x. However, there are two. Review your implicit differentiation skills and use them to solve problems. If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains.kastatic.org and.kasandbox.org are unblocked.
- 6.1 Implicit Vs Explicitap Calculus Algebra
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I want to close this series with a topic that arises constantly, both in classrooms and on social media: How do you evaluate an expression like (adiv bc) or (8div 4(3-1)), where the multiplication is indicated without a specific symbol? There are several reasons one might want to interpret this differently than the rule we’ve discussed, that multiplication and division are done from left to right. We’ll look at this first from the perspective of students and teachers, and then (next time) investigate some historical issues to close out the series.
Two ways to evaluate ax÷by
Let’s first look at one of the earlier questions we had about this issue, in 1999, to set the stage:
(Note that at that time, the only way to type division in our email was to use the slash, (a/b), which I generally assume represents an expression actually written as (adiv b). I will occasionally be inserting an obelus, ÷, where we made rough attempts to simulate it.)
6.1 Implicit Vs Explicitap Calculus Algebra
The first way follows PEMDAS literally, as usually taught and as I’ve presented it here, by evaluating from left to right as (acdot xdiv bcdot y = ((acdot x)div b)cdot y).
The second sees it as (axdiv by = (ax)div (by)). This isn’t explained as following any taught rule, but just as doing what looks right, either because the division is read as if it were a fraction bar, or just because “by” looks like it belongs together as a unit. We’ll be seeing several reasons students have given for doing this.
Though I had been with Ask Dr. Math less than a year, this was already a familiar question, which I wanted to answer thoroughly for the sake of the archive:
Note that it is not only students doing what feels right, but also some textbooks and calculators that follow the second method.
A new rule, or what looks right?
I elaborated on the two methods, taking the PEMDAS version as correct (though I’ll have some second thoughts on that):
I think I had made up the term “implied, or implicit, multiplication” when I answered my first question on the topic a few months before, to refer to multiplication indicated by just putting two numbers or variables or parenthesized expressions next to one another – “juxtaposition“, as others call it – like ( ab) or ( 2b) or ( a(b+c)), as opposed to explicitly writing ( atimes b) or ( acdot b).
We had seen some questions from students whose textbooks taught only the usual PEMDAS, yet evaluated the second way in examples or solutions, without comment. This might have been due to the answers in the back being written by someone other than the author, but it is an inexcusable inconsistency.
Why would an author make this extra rule? I have had different opinions at various times about whether the rule is a good idea, but have always recognized that it is not what is usually taught:
A rule that is not a rule is worthless, no matter how reasonable it is. Yes, the “new rule” is the natural way to read (axdiv by) because (by) looks like a single entity; but until everyone teaches that, we can’t do it and expect to be understood by all readers.
In particular, many students assume that it represents a horizontal version of (displaystylefrac{ax}{by}):
By using parentheses, we can avoid writing something that people who were taught different rules, or who ignore the rules they were taught, might take differently than we intend.
Calculator issues
The link there went bad long ago; but when a specific question about a calculator came up in 2008, I quoted from what TI said in their Knowledge Base:
This makes it clear that calculator designers have to decide on their own rules, which don’t have to be the same as rules for writing on paper; but educators seem to have convinced them to keep things as much the same as possible for students’ sake.
In conclusion (back to the 1999 answer):
Subsequently, we had many more questions about this; I’ll just quote a few unique bits from some of those answers.
Old-fashioned math?
Here is a typical example of a school conflict, from 2000:
The problem is (Ndiv ml), and the parents are doing the multiplication first. I replied, in part:
I explained the standard rules, and added:
I’ll have more on the history next time.
More recently, the fights tend to be on social media!
Misapplying the distributive property
I’ll close with the most recent archived discussion. This question is from 2017:
I answered with a collection of my standard answers to this sort of question; even my first archived answer on the topic in 1999 was largely a standard response I had given to others before. Here, I’ll just look at a few points I made that haven’t been fully covered above.
I first summarized what was going on:
If you recall earlier statements that PEMDAS is (a) in harmony with the properties of operations, and (b) fitting with the visual impression of our notation, then some alarm bells should already be going off!
It is not really the distributive property that led to the “wrong” result, but the fact that in distributing, the 4 was seen as the multiplier.
I have observed that many students learn to distribute so well that they automatically do it when it is not helpful, as here, and even when it is not applicable, as when there is no addition at all!
Why students do it
In an unarchived answer in 2012, I listed reasons students have given for doing the multiplication first; on this occasion the question was about the expression (a^2div 4b + c):
For an example of “sticky parentheses”, see
For an example of seeing the division sign as a fraction bar (and a long discussion of not being swayed by visual appearance), see
Back to the 2017 answer …
Avoidance is the best policy
I closed with a plea for peace:
Arguments on social media about this sort of thing are a waste of time. But thinking about our conventions can be very enlightening. Next time, I’ll close everything out with a look at history, and some solid reasons to think the “new rule” is in fact correct.
Integration Mini Video Lecture
This video explains the differences and similarities between indefinitie and definite integration.
Video transcript
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Related lessons on IntMath
This mini video lecture explains the processes in these two pages:
... and ...
What's the difference between indefinite and definite integrals?
Indefinite integral
With an indefinite integral there are no upper and lower limits on the integral here, and what we'll get is an answer that still has x's in it and will also have a K, plus K, in it.
A definite integral has upper and lower limits on the integrals, and it's called definite because, at the end of the problem, we have a number - it is a definite answer.
OK. Let's do both of them and see the difference.
The integral x to the five dx is equal to ... `(int x^5 dx =?)`
... Now, for integration, I have to add one to the index.
So, it's going to be x to the 6
...and then I divide by the new number ... x to the sixth over 6 `(int x^5 dx = x^6/6...)`
...and then I must remember the plus K. `(int x^5 dx = x^6/6+K)`
K is just some constant. We might have some information elsewhere in the problem that will help us to find the constant. In this case we don't have any extra information, so it's just plus K.
Definite integral
Now, let's see what it looks like as a definite integral, this time with upper and lower limits, and we'll see what happens. `(int_1^2 x^5 dx = ?)`
Step 1 is to do what we just did. We integrate, and I'm going to have once again x to the six over 6, but this time I do not have plus K - I don't need it, so I don't have it.
And I write it like this,
... and then I have a lower limit of 1
... and an upper limit of 2. `(int_1^2 x^5 dx = [x^6/6]_1^2)`
These are the numbers that come from here in the question - 1 comes from here, 2 comes from here.
So, what I've done is I've integrated, and ... written in the upper and lower limits correctly.
The next step is to actually substitute these upper and lower limits.
So, what I can do is like this ... and like this ... and I'm going to substitute 2 into this position, and 1 into this position.
So, what have I done? `(int_1^2 x^5 dx = [x^6/6]_1^2=2^6/6-1^6/6)`
I've found the integral, and I've substituted 2 into this position ... and 1 into this position ... and I'm going to subtract the difference ... and I'm just going to use Scientific Notebook now to give me that, as a decimal.
And it turns out to be 10.5.
6.1 Implicit Vs Explicitap Calculus Calculator
`int_1^2 x^5 dx = [x^6/6]_1^2` `=2^6/6-1^6/6` `=10.5`
And this is a definite answer.
6.1 Implicit Vs Explicitap Calculus 2nd Edition
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