Results 1 to 8 of 8

Thread: [RESOLVED] Unbelievable confusion in calculating the derivative of a function

  1. #1

    Thread Starter
    Fanatic Member
    Join Date
    Mar 2010
    Posts
    759

    Resolved [RESOLVED] Unbelievable confusion in calculating the derivative of a function

    Hello everyone.
    Let's say we have a function as follows:
    f(x) = 3x2 - 2x + 7

    The goal is to find f '(5x).
    I try to calculate that in two different ways:

    Method 1:
    f '(x) = 6x - 2
    f '(5x) = 6(5x) - 2
    f '(5x) = 30x - 2

    Method 2:
    f(x) = 3x2 - 2x + 7
    f(5x) = 3(5x)2 - 2(5x) + 7
    f(5x) = 75x2 - 10x + 7
    f '(5x) = 150x - 10

    That is two completely different results for f '(5x) !!!!!!

    How can that be?
    If you solve a problem in two (or many) different ways, the result should be the same.
    So, why are the results different in this case?

    Which one is correct?
    And what is wrong in the line of reasoning that leads to the other?

    Please help.
    Thanks.

  2. #2
    PowerPoster
    Join Date
    Nov 2017
    Posts
    3,090

    Re: Unbelievable confusion in calculating the derivative of a function

    Meh, what you've asked is basically the same question as to why the below produces different results:

    Let f(x) = 3x^2 - 2x + 7
    Calculate f'(2).

    Method 1:
    f(x) = 3x^2 - 2x + 7
    f'(x) = 6x - 2
    f'(2) = 6*2 - 2
    f'(2) = 10

    Method 2:
    f(x) = 3x^2 - 2x + 7
    f(2) = 3*(2^2) - 2*2 + 7
    f(2) = 12 - 4 + 7
    f(2) = 15
    f'(2) = 0


    Order matters.

  3. #3
    PowerPoster
    Join Date
    Nov 2017
    Posts
    3,090

    Re: Unbelievable confusion in calculating the derivative of a function

    Here's another example that hopefully reinforces things:

    Let f(x) = x
    Find f'(sin(x))

    Method 1:
    f(x) = x
    f'(x) = 1
    f'(sin(x)) = 1

    Method 2:
    f(x) = x
    f(sin(x)) = sin(x)
    f'(sin(x)) = cos(x)


    Which one do you think is correct?

    Basically, the question posed is, what is the slope of f(x) at sin(x)? And obviously, since f(x) = x, the slope is 1 for any value given, regardless of how that value is arrived at, whether it is a constant or the result of evaluating an additional function of x.

  4. #4
    Fanatic Member
    Join Date
    Jan 2015
    Posts
    596

    Re: Unbelievable confusion in calculating the derivative of a function

    I am not as good in math,
    Just ask ChatGPT

  5. #5
    PowerPoster
    Join Date
    Nov 2017
    Posts
    3,090

    Re: Unbelievable confusion in calculating the derivative of a function

    Or let a = 5x.

    We can confidently say that:
    f(a) = 3a^2 - 2a + 7
    f'(a) = 6a - 2

    Substitute 5x for a:
    f'(5x) = 6*5x - 2
    f'(5x) = 30x - 2

  6. #6

    Thread Starter
    Fanatic Member
    Join Date
    Mar 2010
    Posts
    759

    Re: Unbelievable confusion in calculating the derivative of a function

    Quote Originally Posted by OptionBase1 View Post
    ......

    Order matters.
    Can't be said better than that.

  7. #7

    Thread Starter
    Fanatic Member
    Join Date
    Mar 2010
    Posts
    759

    Re: Unbelievable confusion in calculating the derivative of a function

    Quote Originally Posted by OptionBase1 View Post
    Here's another example that hopefully reinforces things:

    Let f(x) = x
    Find f'(sin(x))

    Method 1:
    f(x) = x
    f'(x) = 1
    f'(sin(x)) = 1

    Method 2:
    f(x) = x
    f(sin(x)) = sin(x)
    f'(sin(x)) = cos(x)


    Which one do you think is correct?

    Basically, the question posed is, what is the slope of f(x) at sin(x)? And obviously, since f(x) = x, the slope is 1 for any value given, regardless of how that value is arrived at, whether it is a constant or the result of evaluating an additional function of x.
    Excellent analogy.
    That certainly acts as proof that method 1 is correct and method 2 is wrong.

  8. #8

    Thread Starter
    Fanatic Member
    Join Date
    Mar 2010
    Posts
    759

    Re: Unbelievable confusion in calculating the derivative of a function

    Quote Originally Posted by OptionBase1 View Post
    Or let a = 5x.

    We can confidently say that:
    f(a) = 3a^2 - 2a + 7
    f'(a) = 6a - 2

    Substitute 5x for a:
    f'(5x) = 6*5x - 2
    f'(5x) = 30x - 2
    Another excellent response.
    It is actually a different approach towards the problem based on a different perspective of looking at the situation, and leads to the same conclusion that the derivative must be calculated first and then the value be applied.

    Thanks a lot for your help.
    Ilia

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •  



Click Here to Expand Forum to Full Width