1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Derivative of 1/x the old fashioned way

  1. Jun 16, 2011 #1
    1. The problem statement, all variables and given/known data

    f(x) = 1/x

    Find the derivative

    2. Relevant equations

    f'(x)= lim h->0

    [f(x+h) - f(x)] / [h]

    3. The attempt at a solution

    I MUST show work and use that formula to arrive at the derivative (1/x^2)

    I can't figure out the algebra.

    I rearrange it countless ways..

    [1/(x+h) - 1/x] - (1/x)] / [h]


    But I can NEVER find a way to algebraically arrive at the derivative! Ever!

    What would you guys do to algebraically solve this??
  2. jcsd
  3. Jun 16, 2011 #2
    You must calculate

    [tex]\lim_{h\rightarrow 0}{\frac{\frac{1}{x+h}-\frac{1}{x}}{h}}[/tex]

    First simplify it (to the point where there's only one fraction, instead of three fractions).
  4. Jun 16, 2011 #3


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    Rearranging your limit expression is not necessarily going to help unless you can do the subtraction [1/(x+h) - 1/x] correctly. Show what you get for this calculation.
  5. Jun 16, 2011 #4
    That gets me no where.



    -h / (x^2+hx)
  6. Jun 16, 2011 #5
    Yes it will :smile:

    And now you still need to divide through h...
  7. Jun 16, 2011 #6


    Staff: Mentor

    This is not the complete quotient. This is just the simplification of the numerator, 1/x - 1/(x + h)

    Dividing by h is equivalent to multiplying by 1/h.
  8. Jun 16, 2011 #7
    That's where I'm stuck.

    How can I get rid of the complex fraction here?

    "Dividing through h" doesn't seem possible.
  9. Jun 16, 2011 #8


    Staff: Mentor

    [tex]\lim_{h\rightarrow 0}{\frac{\frac{1}{x+h}-\frac{1}{x}}{h}} = \lim_{h \to 0} \frac{-h}{x^2 + hx}\cdot \frac{1}{h}[/tex]

    Can you simplify a bit more and then take the limit?
  10. Jun 16, 2011 #9
    NOW I see! Thanks!

    -h / hx^2 + h^2x


    h(-1) / h(x^2+hx)=

    (-1) / x^2 + hx

    as h approaches 0, hx approaches zero, leaving me with x^2 in the denomenator for a final result of

    - 1/x^2
  11. Jun 16, 2011 #10


    Staff: Mentor

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook