Derivatives: Solving F'(x) = [2f(x)+g(x)]' Problem

[Mark44]

The value of the derivative at x=2 is f '(2) = 3(2)^2=12.

derivative of what?

You didn't quote all of ehild's comment, which I show below. To be more explicit, it would say "f'(2) is the value of the derivative of f, with respect to x, evaluated at x = 2.

f '(2) is the value of the derivative at x=2. [f(2)] ' is the derivative of the number f(2), which is zero.

Let f be the function defined by f(t)=1-t for all real numbers t.

Is "t"a number?

Of course. "For all real numbers t" means that t is a real number.

Here x&y are functions as f'(x) is function,so x and y are just functions not functions of anything,right?

No. What you wrote is almost completely wrong. x and y are variables that have numeric values. If we are given that y = f(x), then f is a function and f' is another function, the derivative of f with respect to x.

There is no such thing as "just a function" -- a function is always a function of one or possibly more variables. A function always has at least one argument, a value from the domain of the function.

 

[Mister T]

$f'(x)$=$[f(x)]'$=$(x)'$

$f'(x)=[f(x)]'\neq(x)'$.

Let's suppose $y=f(x)$.

$f'(x)$ means the derivative of $y$ with respect to $x$. We can also write this as $\frac{dy}{dx}$. That is, $f'(x)=\frac{dy}{dx}$.

In this context $(x)'=\frac{dx}{dx}=1$. That is, the derivative of $x$ with respect to $x$ equals $1$.

You are mixing up the derivative of x with the derivative of f(x).

In your original post you were mixing up the derivative of 2 with f'(2). The derivative of 2 is zero, f'(2) is the derivative of f(x) evaluated at x = 2.

These mix-ups are the reason why many people prefer the notion $\frac{d}{dx}f(x)$ to $f'(x)$. The latter notation is instead preferred by others because of its brevity. I think it's important to introduce both notations to students right from the start.

 

[Fredrik]

You didn't quote all of ehild's comment, which I show below. To be more explicit, it would say "f'(2) is the value of the derivative of f, with respect to x, evaluated at x = 2.

The phrase "derivative of f with respect to x" is at best an abuse of terminology. The "input" variable in the definition of f is always a dummy variable. For example, if you define f by saying that "f(x)=5x for all real numbers x", then the fact that this statement is equivalent to "f(y)=5y for all real numbers y" implies that the definition hasn't assigned a special role to the variable x.

It's OK to say "derivative of f(x) with respect to x". The "with respect to x" let's the reader know that the writer is talking about the function that takes x to f(x) (i.e. f) rather than, say, the function that takes t to f(x).

There is no such thing as "just a function" -- a function is always a function of one or possibly more variables. A function always has at least one argument, a value from the domain of the function.

If it's been specified that x and y are real numbers such that x+y=1, and we define a function f by f(t)=1-t for all real numbers t, then f is indeed "just" a function, while y and f(x) are functions of x. To say that an expression that involves x is a "function of" x is to say that the value of that expression is determined by the value assigned to x.

 

[Mark44]

In your original post you were mixing up the derivative of 2 with the derivative of f(2). The derivative of 2 is zero, the derivative of f(2) is the derivative of f(x) evaluated at x = 2.

f(2) is a number, so it's derivative is 0.

What I think you meant, as opposed to what you wrote, was f'(2) is the derivative of f, evaluated at x = 2. It's an error to call f'(2) the derivative of f(2).

 

[Mark44]

The phrase "derivative of f with respect to x" is at best an abuse of terminology.

Maybe, or maybe not. I believe my meaning would be clear to virtually all readers with competence in calculus, with the possible exception of the most pedantic of the bunch.

The "with respect to x" part implies that f is a function whose inputs we are calling x values.

There is no such thing as "just a function" -- a function is always a function of one or possibly more variables. A function always has at least one argument, a value from the domain of the function.

If it's been specified that x and y are real numbers such that x+y=1, and we define a function f by f(t)=1-t for all real numbers t, then f is indeed "just" a function, while y and f(x) are functions of x. To say that an expression that involves x is a "function of" x is to say that the value of that expression is determined by the value assigned to x.

As you have defined f above (i.e., "we define a function f by f(t) = 1 - t..." , it is a function of t.

Fredrik, many of your fine points are completely lost on the OP. Your comments would be more helpful to her, IMO, if you geared them toward her level of comprehension.

 

[Fredrik]

As you have defined f above (i.e., "we define a function f by f(t) = 1 - t..." , it is a function of t.

Since the t in the sentence that defines f can be replaced by any other variable without changing the meaning of the sentence, f is no more a function of t than it is a function of z.

Fredrik, many of your fine points are completely lost on the OP. Your comments would be more helpful to her, IMO, if you geared them toward her level of comprehension.

I believe that you are wrong about this. I think that it's even more important to "be pedantic", i.e. use proper terminology, when we're talking to someone who's just learning the terminology.

 

[Mister T]

f(2) is a number, so it's derivative is 0.

What I think you meant, as opposed to what you wrote, was f'(2) is the derivative of f, evaluated at x = 2. It's an error to call f'(2) the derivative of f(2).

You're right. I went back and edited my post to fix my error. Thanks.

 

[Mark44]

As you have defined f above (i.e., "we define a function f by f(t) = 1 - t..." , it is a function of t.

Since the t in the sentence that defines f can be replaced by any other variable without changing the meaning of the sentence, f is no more a function of t than it is a function of z.

I don't buy it. As you wrote it, f is clearly a function of t. If you had defined it as f(z) = 1 - z, I would say that f is a function of z. Of course the letter used is immaterial, but once you provide a formula for the function, you are specifying the independent variable.

Fredrik, many of your fine points are completely lost on the OP. Your comments would be more helpful to her, IMO, if you geared them toward her level of comprehension.

I believe that you are wrong about this. I think that it's even more important to "be pedantic", i.e. use proper terminology, when we're talking to someone who's just learning the terminology.

And I believe that you are wrong. I doubt that gracy understood more than 10% of your explanation in post #27. I'm all in favor of using correct terminology, but if the difference between "correct" terminology and a looser explanation is miniscule, and the beginning student is more able to absorb the looser explanation, that's the side I will err on.

As far as being pedentic, here's an example from this thread.

Is "t"a number?

That's a surprisingly difficult question to answer. When I'm looking at the expression 1-t or the equation f(t)=1-t, I'm inclined to say that t is a number (because the minus sign refers to subtraction of real numbers, and f only takes real numbers as input). But when I'm looking at the complete sentence, I'm inclined to say that since the variable t is never assigned a value, it would be weird to describe it as anything more than a variable.

This is way more subtlety than the OP is capable of understanding, based on her questions in this thread.

 

[Fredrik]

I don't buy it. As you wrote it, f is clearly a function of t. If you had defined it as f(z) = 1 - z, I would say that f is a function of z. Of course the letter used is immaterial, but once you provide a formula for the function, you are specifying the independent variable.

Define f and g by f(x)=1-x for all real numbers x and g(y)=1-y for all real numbers y. You understand that f=g, right? Since the variables f and g represent the same function, it makes no sense to say that f is a function of x, and g is a function of y. Everything that f is, g is too.

I doubt that gracy understood more than 10% of your explanation in post #27.
[...]
This is way more subtlety than the OP is capable of understanding,

You don't know what she's capable of. It's far more reasonable to assume that she's asking questions because she wants to understand the subtleties than to assume that she's asking because she's not capable of understanding them.

One of the things she asked about is why f'(x)=[f(x)]' doesn't imply f'(2)=[f(2)]'. It's impossible to understand this without understanding some of the subtleties. In particular, one needs to understand why the notation on the right-hand sides is absurd.

 

[Mark44]

I don't buy it. As you wrote it, f is clearly a function of t. If you had defined it as f(z) = 1 - z, I would say that f is a function of z. Of course the letter used is immaterial, but once you provide a formula for the function, you are specifying the independent variable.

Define f and g by f(x)=1-x for all real numbers x and g(y)=1-y for all real numbers y. You understand that f=g, right? Since the variables f and g represent the same function, it makes no sense to say that f is a function of x, and g is a function of y. Everything that f is, g is too.

Yes, of course I understand that f and g represent the same function. Once you have said, "Define f by f(x) = 1 - x", then f is a function of x. Similar for your definition of g as a function of y.

I doubt that gracy understood more than 10% of your explanation in post #27.
[...]
This is way more subtlety than the OP is capable of understanding,

You don't know what she's capable of. It's far more reasonable to assume that she's asking questions because she wants to understand the subtleties than to assume that she's asking because she's not capable of understanding them.

Actually, I do have a good idea of what she's capable of, based on seeing her posts in a number of other threads. In this and other threads, there are subtleties that went right past her, resulting in some very long threads.

One of the things she asked about is why f'(x)=[f(x)]' doesn't imply f'(2)=[f(2)]'. It's impossible to understand this without understanding some of the subtleties.
In particular, one needs to understand why the notation on the right-hand sides is absurd.

I disagree. If one understands what the notation means, which gracy clearly does not (refer to her confusion about whether f'(x) = x'), the question is very straightforward. As far as [f(2)]' is concerned, I believe that most people who are competent at calculus would read this as "the derivative, with respect to whatever variable f is a function of, of the number f(2)." IOW, as the derivative of a constant.

 

[gracy]

Fredrik, many of your fine points are completely lost on the OP. Your comments would be more helpful to her, IMO, if you geared them toward her level of comprehension.

I doubt that gracy understood more than 10% of your explanation in post #27.

Actually, I do have a good idea of what she's capable of, based on seeing her posts in a number of other threads. In this and other threads, there are subtleties that went right past her, resulting in some very long threads.

I disagree. If one understands what the notation means, which gracy clearly does not

@Mark44 If you don't want to help.Please don't help.These lines hurt My self esteem .

 

[Mark44]

@Mark44 If you don't want to help.Please don't help.These lines hurt My self esteem .

If I didn't want to help, I wouldn't have posted in posts 2, 11, 14, 36, 39, 40, 43, and 45.

If you want to help with your self esteem, you could start by actually doing the work of this problem, much of which has been done for you by the members of this forum.

 

[gracy]

40, 43, and 45.

Sorry, but These are just replies by you.

2, 11, 14, 36, 39,

YES,you were a great help there.

If you want to help with your self esteem, you could start by actually doing the work of this problem, much of which has been done for you by the members of this forum.

I am not asking problems /numerical questions of calculus here.Just asking for clearer definitions of all the terms so that later I would not have to regret.Strong foundation you know!

I will try to understand each and every line.It will take time.Calculus is very new to me.

Here I have clearly mentioned my intentions .
Please ,don't comment on my capabilities.It's very discouraging .I am a newbie ,my age is less than age of your teaching profession,that's why may be things which are easier for you can be hardest for me.It doesn't mean I am not capable of anything ,that's how I am encouraging myself now !

 

[Fredrik]

Yes, of course I understand that f and g represent the same function. Once you have said, "Define f by f(x) = 1 - x", then f is a function of x. Similar for your definition of g as a function of y.

So f and g are the same, but f is something that g isn't?

Since f=g, I think it makes more sense to say this: f and g are functions. f(x) and g(x) are functions of x. f(y) and g(y) are functions of y. f(r) and g(r) are functions of r.

You seem to be thinking of functions as constraints on value assignments, rather than as mathematical objects that satisfy the definition of "function". It's not a bad way to think of them. It focuses on the idea behind the definition instead of on the definition itself. If that's how you like to think of them, then I would suggest the following terminology:

Let x and y be real numbers such that y=1-x. Now y is a function of x in the sense that the value of y is determined by the value of x. To indicate that, we write y(x) instead of y. So now y and y(x) mean the same thing. That thing isn't a function. It's a real number. What real number it is depends on what real number x represents. Now we haven't chosen a notation for the function, but since the equation y(x)=1-x is what constrains the value assignments, the string of text "y(x)=1-x" can be thought of as a notation for the function.

I believe that most people who are competent at calculus would read this as "the derivative, with respect to whatever variable f is a function of, of the number f(2)." IOW, as the derivative of a constant.

I agree, but the homework forum is mainly for people who aren't yet competent at calculus.

 

[Mister T]

Let x and y be real numbers such that y=1-x. Now y is a function of x in the sense that the value of y is determined by the value of x. To indicate that, we write y(x) instead of y. So now y and y(x) mean the same thing. That thing isn't a function. It's a real number.

The first time I can remember being puzzled by this notation conundrum was when I was taking an undergrad course in quantum mechanics. The function $\Psi$ was being introduced and the professor wrote things like this on the board,

$\Psi=\Psi(x)$,

while saying "Psi is a function of x."

It wasn't until years later that a professor in a graduate math class explained to me the subtleties of that abomination. At the time, though, it was for me just another layer of confusion, piled on top of the other conceptual difficulties associated with quantum mechanics and the meaning of $\Psi$.

Now, as an instructor, I avoid that usage sedulously. We see it used in introductory physics textbooks, particularly the calculus-based variety, in cases like the treatment of simple harmonic motion. The position is x and to make clear that it's a function of the time t the author will write

$x(t)=A \cos(\omega t + \phi)$.

And then go on to refer to x as both that function and the value of that function as if those two things are the same.

Things like this can be a source of confusion for students, and even though the confusion can be easily cleared up it instead stays buried in the student's mind because the confusion itself cannot be articulated by the student.

The OP of this thread, Gracy, was simply trying to get help because she's confused about something. That confusion might be very similar to mine. Unlike me, though, she has the courage to ask. We have an opportunity here to encourage that type of behavior.

One thing I learned when I started teaching is that we see only one facet of a student's ability when we try to teach him a single subject. We don't see the other facets because the student is not showing them to us. I had assigned a project. A list of numerical analysis problems was given to the students. They had to choose one and use spreadsheet software to solve it. One student came to me expressing a curiosity about one of them. It was the most challenging one on the list. Inwardly I sighed because I had formed an impression of this student that led me to believe that that particular problem was beyond her abilities. I steered her towards a simpler one. Months later I learned that that student has an IQ that placed her in the genius category. I still wonder how differently things could have turned out if I had instead helped the student to do the more difficult problem.

So now I try to be patient with students and not pre-judge their abilities. Many students who seem incapable of learning subjects like physics and math are presenting a facet to us that we don't understand. They think about things differently than we do, so there is this impedance mismatch where each of us doesn't "get" the other.

Or as Sheila Tobias famously wrote, "They're Not Dumb, They're Just Different".

 

[Fredrik]

The first time I can remember being puzzled by this notation conundrum was when I was taking an undergrad course in quantum mechanics. The function $\Psi$ was being introduced and the professor wrote things like this on the board,

$\Psi=\Psi(x)$,

while saying "Psi is a function of x."

My teacher for the introductory courses on classical mechanics did the same thing.

It wasn't until years later that a professor in a graduate math class explained to me the subtleties of that abomination.

I actually tried to use it on a math exam. The professor dismissed it by saying that you can't write down an equality where one side is a function and the other isn't. I tried to explain what I was doing there, but he wasn't even interested in hearing it. I had to figure it out on my own after that. I have always been careful with the notation and the terminology since then.

At the time, though, it was for me just another layer of confusion, piled on top of the other conceptual difficulties associated with quantum mechanics and the meaning of $\Psi$.

Now, as an instructor, I avoid that usage sedulously.
[...]
Things like this can be a source of confusion for students, and even though the confusion can be easily cleared up it instead stays buried in the student's mind because the confusion itself cannot be articulated by the student.

Yes, that's what I'm thinking too. Thanks for posting.

 

[Mark44]

The first time I can remember being puzzled by this notation conundrum was when I was taking an undergrad course in quantum mechanics. The function $\Psi$ was being introduced and the professor wrote things like this on the board,

$\Psi=\Psi(x)$,

while saying "Psi is a function of x."

It wasn't until years later that a professor in a graduate math class explained to me the subtleties of that abomination. At the time, though, it was for me just another layer of confusion, piled on top of the other conceptual difficulties associated with quantum mechanics and the meaning of $\Psi$.

Now, as an instructor, I avoid that usage sedulously. We see it used in introductory physics textbooks, particularly the calculus-based variety, in cases like the treatment of simple harmonic motion.

So it would seem that Fredrik's and your purpose in life is to get all of the authors of physics textbooks to stop using this notation? Good luck with that. I am less concerned about an abuse of notation than I am about being able to communicate an idea. Above, the dependent variable $\Psi$ is some function of x. Perhaps it would be better to write $\Psi = f(x)$, but as long as the idea of connectedness between $\Psi$ and x is communicated, I'm OK with the notation.

I do agree, though, that it would be awful if math texts wrote something like "Let f = f(x) = 2x + 3".

The position is x and to make clear that it's a function of the time t the author will write

$x(t)=A \cos(\omega t + \phi)$.

Is there any evidence that this actually causes confusion?

And then go on to refer to x as both that function and the value of that function as if those two things are the same.

Things like this can be a source of confusion for students, and even though the confusion can be easily cleared up it instead stays buried in the student's mind because the confusion itself cannot be articulated by the student.

 

[Dale]

Closed pending moderation.