How do I compute the following differentiation by chain rule?

[gentsagree]

How do I compute the following differentiation by chain rule?

\frac{d}{d\lambda}(\lambda^{-1}\phi(\lambda^{-1}x))

It is not a homework, but I can't figure out the exact way of getting the answer -\phi(x)-x^{s}\partial_{s}\phi(x)

 

[HallsofIvy]

First, this is a product, \lambda^{-1} times \phi(\lambda^{-1}x) so you have to use the product rule. The derivative of \lambda^{-1} with respect to \lambda is -\lambda^{-2} and the derivative of \phi(\lambda^{-1}x) with respect to \lambda is \phi'(\lambda^{-1}x) (-\lambda^{-2}x).

So the derivative of \lambda^{-1}\phi(\lambda^{-1}x) is -\lambda^{-2}\phi(\lambda^{-1}x)- \lambda^{-3}x\phi'(\lambda^{-1}x). That is NOT what you give but I don't know what -\phi(x)- x^s \partial_s\phi(x) means because there is no "s" in your statement of the problem. Nor do I know where the "\lambda" disappeared to! Are you sure you have stated the problem correctly?

Is it possible that this is the derivative at \lambda= 1? But I still don't understand what "s" is.