Results 1 to 5 of 5
Like Tree2Thanks
  • 1 Post By chiro
  • 1 Post By hollywood

Math Help - Please help me understand Leibniz' notation

  1. #1
    Senior Member Paze's Avatar
    Joined
    Nov 2012
    From
    Iceland
    Posts
    379
    Thanks
    19

    Please help me understand Leibniz' notation

    For some inexplicable reason, I cannot seem to understand Leibniz' notation.

    I get it at its most simple form such as: \frac{dy}{dx}x^2 I do see that we are taking the difference (or delta) of y which is \left( (x+\Delta x)(x+\Delta x)-x^2\left) and then I take the difference (delta) of x which is x+\Delta x-x and then of course, take the limit as \Delta x \rightarrow 0

    However when we start playing around with Leibniz' notation, I am at a loss. For instance, now I am trying to understand this

    Let w=\ln

    \frac{d}{dx}e^w=\frac{d}{dx}x=1\\\\\Rightarrow \left(\frac{d}{dw}e^w\right)\left(\frac{dw}{dx} \right)=1\\\\\Rightarrow e^w\cdot \frac{dw}{dx}=1\Rightarrow\frac{dw}{dx}=\frac{1}{e  ^w}=\frac{1}{x}

    (if it helps, here are my notation for the lecture: http://i.imgur.com/1fjBycY.jpg. The problem is on the right side.)

    But I am having severe issues understanding what is going on!

    I do understand the calculations if presented to me without the Leibniz' notation, I am sure. But this notation seems to clog me up every time! Please help MHF!
    Last edited by Paze; February 17th 2013 at 08:44 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,619
    Thanks
    592

    Re: Please help me understand Leibniz' notation

    Hey Paze.

    Basically this is just the chain rule where for w = f(x) and g(w) = e^w and you are differentiating d/dx g(f(x)).
    Thanks from Paze
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member Paze's Avatar
    Joined
    Nov 2012
    From
    Iceland
    Posts
    379
    Thanks
    19

    Re: Please help me understand Leibniz' notation

    Quote Originally Posted by chiro View Post
    Hey Paze.

    Basically this is just the chain rule where for w = f(x) and g(w) = e^w and you are differentiating d/dx g(f(x)).
    Thank you, I did suspect that but my problem is understanding what is going on with the notation. Why does \frac{d}{dx} become \frac{d}{dw}?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Mar 2010
    Posts
    980
    Thanks
    236

    Re: Please help me understand Leibniz' notation

    Don't let Leibniz's notation scare you. Whether you write \frac{d}{dx}f(x) or f'(x), it's just the derivative of f.

    Your professor introduced a new variable (or a new function, depending on how you look at it) w=\ln{x}. So e^w=x, and you take the derivative of both sides. The right side is easy. On the left, since w is a function of x, you have a composite function ( e^w is e^{\ln{x}}), so you need to use the chain rule. The derivative of the outside function is \frac{d}{dw}e^w=e^w, and the derivative of the inside function is \frac{dw}{dx}. So you have e^w\frac{dw}{dx}=1, so \frac{dw}{dx}=\frac{1}{e^w}=\frac{1}{x}, and since w=\ln{x}, \frac{d}{dx}\ln{x}=\frac{1}{x}, which is what you wanted to show.

    If you can't figure out the derivation, you should at least memorize the result \frac{d}{dx}\ln{x}=\frac{1}{x}. Or in the other notation, if f(x)=\ln{x}, f'(x)=\frac{1}{x}.

    Hope that helps.

    - Hollywood
    Thanks from Paze
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member Paze's Avatar
    Joined
    Nov 2012
    From
    Iceland
    Posts
    379
    Thanks
    19

    Re: Please help me understand Leibniz' notation

    Quote Originally Posted by hollywood View Post
    Don't let Leibniz's notation scare you. Whether you write \frac{d}{dx}f(x) or f'(x), it's just the derivative of f.

    Your professor introduced a new variable (or a new function, depending on how you look at it) w=\ln{x}. So e^w=x, and you take the derivative of both sides. The right side is easy. On the left, since w is a function of x, you have a composite function ( e^w is e^{\ln{x}}), so you need to use the chain rule. The derivative of the outside function is \frac{d}{dw}e^w=e^w, and the derivative of the inside function is \frac{dw}{dx}. So you have e^w\frac{dw}{dx}=1, so \frac{dw}{dx}=\frac{1}{e^w}=\frac{1}{x}, and since w=\ln{x}, \frac{d}{dx}\ln{x}=\frac{1}{x}, which is what you wanted to show.

    If you can't figure out the derivation, you should at least memorize the result \frac{d}{dx}\ln{x}=\frac{1}{x}. Or in the other notation, if f(x)=\ln{x}, f'(x)=\frac{1}{x}.

    Hope that helps.

    - Hollywood
    I get it!

    God I love when the solution just appears to you after scouring for minutes, hours or even days. What you wrote was Hebrew to me for the first 30 minutes or so but after that it just suddenly clicked. That feeling is just #1. Thank you very much.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. leibniz notation
    Posted in the Calculus Forum
    Replies: 5
    Last Post: April 23rd 2011, 07:51 AM
  2. Replies: 1
    Last Post: March 10th 2011, 02:23 AM
  3. Leibniz notation.
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 9th 2010, 04:20 AM
  4. Chain Rule Leibniz Notation
    Posted in the Calculus Forum
    Replies: 5
    Last Post: March 7th 2010, 05:55 PM
  5. I do not understand Leibniz notation
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 21st 2009, 06:26 PM

Search Tags


/mathhelpforum @mathhelpforum