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!

Chain rule

  1. Apr 27, 2015 #1
    1. The problem statement, all variables and given/known data
    Let φ=φ(r) and r=√x2+y2+z2. Find ∂2φ/∂x2.
    Show that it can be written as (1/r + x2/r3)∂φ/∂r + x2/r22φ/∂r2.
    2. Relevant equations
    Use the identity ∂r/∂x = x/r.

    3. The attempt at a solution
    I think I know ∂φ/∂x. Using the chain rule, it's ∂r/∂x ∂φ/∂r. That gives x/r ∂φ/∂r. If that's wrong it might be because I know you have to take account of all dependences, but I don't actually know how to.
    So assuming that's ok, I then need to use the product rule, and that gave me
    1/r ∂φ/∂r + ∂φ/∂x ∂φ/∂r ∂r/∂x.
    Which I know is wrong, because it's a show that question!
  2. jcsd
  3. Apr 27, 2015 #2


    Staff: Mentor

    I would write this as dφ/dr ∂r/∂x or φ'(r)∂r/∂x = dφ/dr (x/r). φ is a function of r alone, so the derivative for this function is the regular derivative instead of the partial derivative. Now use the product rule to get ∂2φ/∂x2.
  4. Apr 29, 2015 #3
    I'm doing something wrong. So I got ∂2φ/∂x2 = 1/r ∂φ/∂r + ∂/dr (dφ/dr) ∂r/∂x x/r. This doesn't give me the answer, it gives
    2φ/∂x2 = 1/r dφ/dr + x2/r22φ/∂r2, but I really can't see what's wrong!
  5. Apr 29, 2015 #4
    I'm missing a whole term: x2/r3 dφ/dr.
  6. Apr 29, 2015 #5


    Staff: Mentor

    You don't show the work leading up to this, but I'm guessing that you did this: ##\frac{\partial}{\partial x} \frac{x}{r} = \frac{1}{r}##. If so, that's wrong, since you would be treating r as a constant. In fact, r is a function of x (and y and z).
  7. Apr 29, 2015 #6
    Ohhh... That's exactly what I did. I always forget that!
  8. Apr 29, 2015 #7
    Should be y2+z2/r3 then.
    Yep, that gives the right answer! Thank you, I was getting really confused.
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: Chain rule
  1. Chain Rule! (Replies: 7)

  2. The chain rule (Replies: 2)

  3. Chain rule (Replies: 1)

  4. Chain rule (Replies: 2)

  5. Chain rule (Replies: 22)