Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Question about limit definition of partial derivative

  1. May 17, 2014 #1
    I've seen it written two different ways:

    $$\frac{\partial f}{\partial x} = \lim\limits_{h \rightarrow 0} \frac{f(x + h, y) - f(x,y)}{h}$$

    and

    $$\frac{\partial f}{\partial x} = \lim\limits_{h \rightarrow 0} \frac{f(x_0 + h, y_0) - f(x_0,y_0)}{h}$$

    where the latter evaluates the function at the respective point before plugging it into the definition of the limit. For example, the function ##f(x,y) = \begin{cases}
    \frac{x^2 y^4}{x^4 + 6y^8}, & \text{if }(x,y) \neq (0,0) \\
    0, & \text{if }(x,y) = (0,0)
    \end{cases}##

    I want to determine if ##\frac{\partial f}{\partial x}## is differentiable at ##(0,0)##.

    Using the second limit definition would make showing the existence of ##\frac{\partial f}{\partial x}## considerably easier, since ##y_0## makes the first term in the limit ##0##, and ##f(x_0,y_0)## is defined to be ##0##.

    But using the first definition, we have to evaluate:

    $$\frac{(x+h)^2 y^4}{(x+h)^4 + 6y^8} - \frac{x^2 y^4}{x^4 + 6y^8}$$

    I'm hoping the "real" or at least usable definition is the second one, but which one is the one we're supposed to use in practice to be technically/mathematically correct?
     
  2. jcsd
  3. May 18, 2014 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Question about limit definition of partial derivative
Loading...