# How do I compute the following differentiation by chain rule?

## Main Question or Discussion Point

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.