Results 1 to 4 of 4

Math Help - Derivative Notation Issue

  1. #1
    Newbie
    Joined
    Oct 2010
    Posts
    18

    Derivative Notation Issue

    Guys I am sorry if this question has been asked before, but it has been pestering me for long time. It is about the notation of derivatives using 'prime'.

    If f(x) is a function of x, then does f'(a) represent f(a) differentiated with respect to a, or does it represent f(x) differentiated with respect to x where x=a or are they both the same thing?

    Forgive the lack of Latex sophistication.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Mar 2010
    Posts
    715
    Thanks
    2
    Quote Originally Posted by Skyrim View Post
    If f(x) is a function of x, then does f'(a) represent f(a) differentiated with respect to a, or does it represent f(x) differentiated with respect to x where x=a or are they both the same thing?
    If your function is f(x), then f'(x) is the derivative of f(x); and f'(a) is the derivative of f(x) evaluated at x = a. If your function is f(a), however, then f'(a) of course represents the derivative of f(a).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,713
    Thanks
    1472
    In other words, it depends upon whether you are thinking of "a" as a variable or as a specific value of a variable. If your function has been written as a function of x and then you see f'(a), that is the derivative with respect to the variable x, evaluated at x= a.

    Of course, if you think of a as being the variable, and differentiate f(a) with respect to a, you should get exactly the same thing as if you differentiated f(x) with respect to x, then set x= a.

    For example, if the function is f(x)= x^3- sin(x) then f'(x)= 3x^2- cos(x) and, setting x= a, f'(a)= 3a^2- cos(a). On the other hand, if I replace the variable x with the variable a, I would have f(a)= a^3- sin(a) and the derivative with respect to a is f'(a)= 3a^2- cos(a), exactly as before.

    Of course, that has to be a letter or something that we can think of as a variable. If we have f(x)= x^3- sin(x) as before and are asked to find f'(\pi), we would set f'(x)= 3x^2- cos(x) and then set x= \pi: f'(\pi)= 3\pi^2+ 1.

    It would be a terrible mistake to first set x equal to the constant \pi and then argue " f(\pi)= \pi^3- sin(\pi)= \pi^3 is a constant so its derivative is 0". We differentiate with respect to a variable, not a constant.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Oct 2010
    Posts
    18
    Quote Originally Posted by HallsofIvy View Post
    In other words, it depends upon whether you are thinking of "a" as a variable or as a specific value of a variable. If your function has been written as a function of x and then you see f'(a), that is the derivative with respect to the variable x, evaluated at x= a.

    Of course, if you think of a as being the variable, and differentiate f(a) with respect to a, you should get exactly the same thing as if you differentiated f(x) with respect to x, then set x= a.

    For example, if the function is f(x)= x^3- sin(x) then f'(x)= 3x^2- cos(x) and, setting x= a, f'(a)= 3a^2- cos(a). On the other hand, if I replace the variable x with the variable a, I would have f(a)= a^3- sin(a) and the derivative with respect to a is f'(a)= 3a^2- cos(a), exactly as before.

    Of course, that has to be a letter or something that we can think of as a variable. If we have f(x)= x^3- sin(x) as before and are asked to find f'(\pi), we would set f'(x)= 3x^2- cos(x) and then set x= \pi: f'(\pi)= 3\pi^2+ 1.

    It would be a terrible mistake to first set x equal to the constant \pi and then argue " f(\pi)= \pi^3- sin(\pi)= \pi^3 is a constant so its derivative is 0". We differentiate with respect to a variable, not a constant.
    Thank you so much.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. nth Derivative notation question
    Posted in the Calculus Forum
    Replies: 6
    Last Post: August 8th 2011, 04:52 PM
  2. Mathcad Second Derivative Test Issue
    Posted in the Math Software Forum
    Replies: 3
    Last Post: March 21st 2011, 11:35 AM
  3. Replies: 1
    Last Post: March 10th 2011, 02:23 AM
  4. Replies: 11
    Last Post: December 5th 2009, 05:44 PM
  5. Derivative issue
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 4th 2008, 03:03 PM

Search Tags


/mathhelpforum @mathhelpforum