Forced to use the chain rule

[Oaky]

Okay, so essentially this is the second part of a question, and I'm actually finding the second derivative after already getting the first, but I'll just ignore the fact that this is actually the second to make things clearer.

I need to find the derivative of \(\displaystyle \displaystyle{-x \over (x^2+y^2+z^2)^{3/2}}\), and have been told to use the chain rule with \(\displaystyle r=\sqrt{x^2+y^2+z^2}\).

So is there any way to utilise the chain rule with \(\displaystyle \displaystyle{-x \over r^3}\), because it seems as though I'd need everything in terms of r first. The question could very well have been asking to use the chain rule for the first derivative only, but there might be a way to somehow use the r value in this part of the question.

Thanks!

 

[biffboy]

Okay, so essentially this is the second part of a question, and I'm actually finding the second derivative after already getting the first, but I'll just ignore the fact that this is actually the second to make things clearer.

I need to find the derivative of \(\displaystyle \displaystyle{-x \over (x^2+y^2+z^2)^{3/2}}\), and have been told to use the chain rule with \(\displaystyle r=\sqrt{x^2+y^2+z^2}\).

So is there any way to utilise the chain rule with \(\displaystyle \displaystyle{-x \over r^3}\), because it seems as though I'd need everything in terms of r first. The question could very well have been asking to use the chain rule for the first derivative only, but there might be a way to somehow use the r value in this part of the question.

Thanks!

It would help to know what the first part of the question said.

 

[Oaky]

Okay, both questions:

2. Show that the function \(\displaystyle u(x,y,z)\) is a solution of the partial differential equation \(\displaystyle u_{xx} + u_{yy} + u_{zz} = 0\) (this is known as Laplace's Equation).

3. Redo the previous problem using the chain rule, setting \(\displaystyle r = \sqrt{x^2+y^2+z^2}\).


So I can use the chain rule with that value of r to find \(\displaystyle u_x\), but essentially I'm wondering if you can use it again to find \(\displaystyle u_{xx}\), or if you should just use the standard quotient rule (with an easier "do in your head" chain rule included in that).

 

[Deleted member 4993]

Okay, both questions:

2. Show that the function \(\displaystyle u(x,y,z)\) is a solution of the partial differential equation \(\displaystyle u_{xx} + u_{yy} + u_{zz} = 0\) (this is known as Laplace's Equation).

3. Redo the previous problem using the chain rule, setting \(\displaystyle r = \sqrt{x^2+y^2+z^2}\).


So I can use the chain rule with that value of r to find \(\displaystyle u_x\), but essentially I'm wondering if you can use it again to find \(\displaystyle u_{xx}\), or if you should just use the standard quotient rule (with an easier "do in your head" chain rule included in that).

Where is your u(x,y,z) ?

 

[Oaky]

Shite sorry, the LaTeX for that part never appeared.

\(\displaystyle u(x,y,z) =\displaystyle \frac{1}{\sqrt{x^2+y^2+z^2}}\)

 

[daon2]

Shite sorry, the LaTeX for that part never appeared.

\(\displaystyle u(x,y,z) =\displaystyle \frac{1}{\sqrt{x^2+y^2+z^2}}\)

Okay, so \(\displaystyle r = \sqrt{x^2+y^2+z^2}\)

You have \(\displaystyle u_x = -x\cdot r^{-3}\)

The product rule gives \(\displaystyle u_{xx} = 3x\cdot r^{-4}\cdot \frac{dr}{dx}\, -\,r^{-3}\)