# Calculating the derivative of a mapping $\varphi: S^2 \rightarrow S^2$

I have recently learned about tangent spaces and derivatives in the context of manifolds and I am having a hard time solving the following exercise:

Let A be a $$3\times3$$ orthogonal matrix. Consider the map $$\varphi: S^2 \rightarrow S^2$$ defined by $$\varphi(x) = Ax$$.
($$S^2$$ means the 2 sphere)

Calculate the mapping $$D\varphi(x) = T_x S^2 \rightarrow T_{\varphi(x)}S^2$$

If I am not mistaking I need to find bases in $$T_x S^2$$ and $$T_{\varphi(x)}S^2$$ and then calculate $$D(f_2^{-1}\circ\varphi \circ f_1)(x)$$ where $$f_1$$ and $$f_2$$ are local charts but I do not know how to do this in practice.

Could you help me?

$$S^2$$ is a smooth submanifold of $$\mathbb{R}^3$$. Let $$i : S^2 \to \mathbb{R}^3$$ denote the (smooth) embedding and let $$\phi : \mathbb{R}^3 \to \mathbb{R}^3, \phi(x) = Ax$$. Then $$(*) \phantom{x} Di(\varphi(x)) D\varphi(x) = D\phi(x) Di(x) .$$ But $$T_y \mathbb{R}^3 = \mathbb{R}^3$$ and $$D\phi(y) = \phi$$ for all $$y \in \mathbb{R}^3$$. The tangent spaces $$T_yS^2$$, $$y \in S^2$$, can be identified via $$Di(y)$$ with two-dimensional linear subspaces of $$\mathbb{R}^3$$. Doing so, $$(*)$$ shows that $$\phi(T_x S^2) = T_{\varphi(x)} S^2$$ and $$D\varphi(x)$$ is the restriction of $$\phi$$.
Concerning the tangent spaces of a smooth submanifold $$M \subset \mathbb{R}^n$$ also see my answer to The motivation for a tangent space.