I was following a lecture on Tangent Spaces, where I find expressions as:
$$(f\circ\gamma\circ\mu)'(0) = (f\circ\gamma)'(\mu(0)).\mu'$$
And in some other place, I find:
$$((f\circ x^{-1})\circ(x\circ\sigma))'(0) = (x\circ\sigma)^{i'}(0).(\partial_{i}(f\circ x^{-1}))(x(\sigma(0))) $$
Now, I am new to Undergraduate Analysis, and cannot understand how the derivative of the above mentioned operators are happening. So can anyone please explain me how these steps come about, or, provide me with some materials to study these.
Thanks in advance.
First is simple chain rule using for $f\circ\gamma$ and $\mu$. Second look likes same chain rule for multiple variables: if rewrite as $$((f\circ x^{-1})\circ(x\circ\sigma))'(0) = \sum_{i}(\partial_{i}(f\circ x^{-1}))(x(\sigma(0)))(x\circ\sigma)^{i'}(0).$$