Quick Questions on Derivatives

[kald13]

I've hit a snag in my studies, namely something my book labels "Corollary 10.1":

Are there any other functions with the same derivative as $x^2+2=2x$? You should quickly come up with several: $x^2+3$ and [itex x^2-4[/itex] for instance. In fact, $d/dx[x^2+c]=2x$ for any constant c. Are there any other functions, though, with the derivative 2x? Corollary 10.1 says that there are no such functions.

Corollary 10.1:
Suppose $g'(x)=f'(x)$ for all x in some open interval I, then for some constant c, $g(x)=f(x) + c$ for all x in the interval I.

As I read it, the text completely contradicted itself. So where is my understanding broken?

Also, I'm not sure why f'(x)=0 is undefined when x=|x|, since it's defined everywhere else. The absolute value of 0 is still 0, isn't it?

 

[Ray Vickson]

I've hit a snag in my studies, namely something my book labels "Corollary 10.1":

Are there any other functions with the same derivative as $x^2+2=2x$? You should quickly come up with several: $x^2+3$ and [itex x^2-4[/itex] for instance. In fact, $d/dx[x^2+c]=2x$ for any constant c. Are there any other functions, though, with the derivative 2x? Corollary 10.1 says that there are no such functions.

Corollary 10.1:
Suppose $g'(x)=f'(x)$ for all x in some open interval I, then for some constant c, $g(x)=f(x) + c$ for all x in the interval I.

As I read it, the text completely contradicted itself. So where is my understanding broken?

Also, I'm not sure why f'(x)=0 is undefined when x=|x|, since it's defined everywhere else. The absolute value of 0 is still 0, isn't it?

Whys do you say the text contradicts itself? All it says (re-phrasing) is that if h(x) = f(x) - g(x) has h'(x) = 0 for all x in the interval I, then h(x) = constant in I. Where is the contradiction? I cannot figure out what is bothering you about that.

The absolute value function is not differentiable at x = 0. If you don't believe that, just look at the graph of y = |x| near x = 0. Does the graph have a unique tangent line through the point (0,0)?

 

[Mark44]

A couple of points that Ray didn't mention:

Also, I'm not sure why f'(x)=0 is undefined when x=|x|, since it's defined everywhere else. The absolute value of 0 is still 0, isn't it?

"when x = |x|" should be "when f(x) = |x|"

It's true that |0| = 0, but that's not relevant when you're talking about f'(x). For this function, if x < 0, f'(x) = -1, and if x > 0, f'(x) = +1. However, there is no unique tangent when x = 0.

 

[kald13]

The first part seems to be a contradiction because it asks you if you can think of any functions with a derivative equal to 2x, and then lists two examples. It then goes on to ask if there are any other functions with the derivative 2x (after it just gave us three) and says there are none.

Are there any other functions with the derivative 2x? Yes, here are two. Are there any others? No, there are none.

 

[scurty]

I would hazard a guess to say that the book probably meant to say there is no other family (set) of functions other than $f(x) = x^2 + c, \: c \in \mathbb{R}$ that satisify $f'(x) = 2x$.

 

[Mandelbroth]

I've hit a snag in my studies, namely something my book labels "Corollary 10.1":

Are there any other functions with the same derivative as $x^2+2=2x$? You should quickly come up with several: $x^2+3$ and [itex x^2-4[/itex] for instance. In fact, $d/dx[x^2+c]=2x$ for any constant c. Are there any other functions, though, with the derivative 2x? Corollary 10.1 says that there are no such functions.

Corollary 10.1:
Suppose $g'(x)=f'(x)$ for all x in some open interval I, then for some constant c, $g(x)=f(x) + c$ for all x in the interval I.

As I read it, the text completely contradicted itself. So where is my understanding broken?

Also, I'm not sure why f'(x)=0 is undefined when x=|x|, since it's defined everywhere else. The absolute value of 0 is still 0, isn't it?

Because your book went out of its way to make what they said correct, I will refrain from expounding on this too much. However, it's important to note that, when you aren't considering all $x$ on an open interval, the derivatives of two functions being the same only requires that their difference be locally constant.

Germane to the actual subject matter, however, I don't understand the first paragraph you quoted. Is it just saying that $\frac{d}{dx}[x^2+2]=2x$?

 

[HallsofIvy]

I would hazard a guess to say that the book probably meant to say there is no other family (set) of functions other than $f(x) = x^2 + c, \: c \in \mathbb{R}$ that satisify $f'(x) = 2x$.

You don't have to "guess"! The fact that "if f'(x)= g'(x), for all x in some interval (which can be 'all real numbers') then f and g differ only by a constant" follows form the "mean value theorem".

You know that if f'(x)=g'(x) then f'(x)- g'(x)= (f(x)- g(x))'= 0. You know, further, by the "mean value theorem" that if h is differentiable, then for any a and b, (h(b)- h(a))/(b- a)= hf'(c) for some c between a and b. If h'(x)= 0 for all x, then (h(b)- h(a))/(b- a)= 0 so that h(b)= h(a). That is, if h'(x)= 0 for all x in an interval then h is a constant[//b] for all x in that interval.

And that means that if f'(x)= g'(x) for all x in an interval, then f(x)= g(x)+ C, for some constant C, on that interval.