My function is $\phi(x+\lambda a)$, where $x$ and $\lambda$ are variables and $a$ is a constant. I want to calculate what $\frac{\partial\phi}{\partial\lambda}$ is.
Let's define $u=x+\lambda a$ so $\frac{\partial\phi}{\partial\lambda} = \frac{\partial\phi}{\partial u}\frac{\partial u}{\partial\lambda} = a\frac{d\phi}{d u}$
But now I somehow want to change back to an expression in terms of $x$ and $\lambda$.
I do know that $\frac{\partial\phi}{\partial\lambda}|_{\lambda=0} = a \frac{\partial\phi}{\partial x}$ but I can't figure out the general expression for any given $\lambda$