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 derivative

  1. Aug 5, 2017 #1

    dyn

    User Avatar

    1. The problem statement, all variables and given/known data
    The question asks to calculate ∂f/∂x for f(x,y,t) = 3x2 + 2xy + y1/2t -5xt where x(t) = t3 and y(t) = 2t5

    2. Relevant equations
    The answer is given as ∂f/∂x = 6x + 2y - 5t

    3. The attempt at a solution
    I'm confused because the answer given seems to treat x,y ,t as independent variables and the answer given is just a partial derivative treating y and t as constant. But really x,y.t are all dependent on each other. Is it even possible to obtain a partial derivative with respect to x in this case ?
     
  2. jcsd
  3. Aug 5, 2017 #2

    andrewkirk

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The information that x(t) = t3 and y(t) = 2t5 is a red herring. When taking a partial derivative of f with respect to x, we look only at the explicit, direct role of x in the formula for f(x,y,t). We ignore any dependencies, and that leads to the quoted result. If we want to take dependencies into account, we take a total derivative, which is written ##\frac{df}{dx}##, rather than the partial derivative ##\frac{\partial f}{\partial x}##.

    The Insight article on partial derivatives gives more background on partial derivatives, and the nature of these distinctions.
     
  4. Aug 5, 2017 #3

    dyn

    User Avatar

    Thanks. And would I be correct that the total derivative is
    df/dx = ∂f/∂t + (∂f/∂x)dx/dt + (∂f/∂y)dy/dt ?
     
  5. Aug 6, 2017 #4

    andrewkirk

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    That's the formula for the total derivative with respect to t.
    The total derivative wrt x uses the same formula, but swaps the role of t and x on the RHS, giving
    $$\frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial t} \frac{d t}{dx}
    + \frac{\partial f}{\partial y} \frac{dy}{dx}$$
     
  6. Aug 6, 2017 #5

    dyn

    User Avatar

    Thanks for your help
     
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 derivative
  1. Partial derivative (Replies: 21)

  2. The partial derivative (Replies: 2)

  3. Partial derivatives (Replies: 11)

  4. Partial Derivatives (Replies: 4)

  5. Partial derivative (Replies: 7)

Loading...