1. Not finding help here? Sign up for a free 30min 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!

Partial differential coefficient

  1. Jan 5, 2017 #1
    1. The problem statement, all variables and given/known data

    The equation is z= e (x*y), the interesting thing is y is function of x too, y = ψ(x)

    Calculate the partial derivative respect to x, and the total derivative.

    2. Relevant equations

    Total differential:

    dz= ∂z/∂x dx + ∂z/∂y dy

    3. The attempt at a solution

    Well, according to the book:

    ∂z/∂x= y * exy

    But I don't agree with this result, because if y is also function of x this term will be different.

    For example if y=x3, z would be = ex4, and the partial derivative is:

    ∂z/∂x = 4 x3 * e x4

    and this is not the same as:

    y * e xy = x3ex4

    What do you think?

    I just want to check if the book is right, this topic (multivariate differential calculus) is so important that I want to understand it correctly.
     
  2. jcsd
  3. Jan 5, 2017 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    The book is right. When they talk about partial derivative they talk about the derivative considering ##e^{yx}## as a function of ##x## and ##y## as independent parameters. What you are computing is the total derivative.
     
  4. Jan 5, 2017 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    If ##z = e^{x\, \psi(x)} ## then there really is no such thing as a "partial derivative" of ##z##; there is only an "ordinary" (1-variable) derivative ##dz/dx##.
    However, that being said, we do have that ##z'(x)## is expressed in terms of the partial derivatives of the function ##e^{xy}## evaluated at ##y = \psi(x)## (and, of course, the derivative ##d \psi(x)/dx## is involved as well).

    To fix your difficulties when ##y = x^3## you need to use the full force of the complete chain rule:
    $$\frac{d}{dx} \left( \left. e^{x y} \right|_{y = x^3} \right) = \left. \frac{\partial e^{x y}}{\partial x} \right|_{y = x^3}
    + \left. \frac{ \partial e^{xy}}{\partial y} \right|_{y = x^3} \: \cdot \frac{d \, x^3}{dx}$$
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Partial differential coefficient
  1. Partial Differentiation (Replies: 11)

  2. Partial differentiation (Replies: 10)

Loading...