Remark:

The curves $\sigma_1, \sigma_2$ and $\sigma_3$ given by

$$\sigma_1(t) = \cos t + i\sin t,\quad \sigma_2(t)= \cos t + j\sin t, \quad \sigma_3(t) = \cos t + k\sin t$$

lie on $S^3$ and pass through the point $1$. Differentiating and setting $t = 0$ confirms that the vectors $i, j, k$ span the tangent space to $S^3$ at $1$. Thus $D\psi(1)i, D\psi(1)j$ and $D\psi(1)k$ span the image of $D\psi(1)$. We leave it to the reader to check, by calculating the derivatives of $\psi\circ \sigma_j$ ( $j = 1, 2, 3$) at $t = 0$, that $D\psi(1)i, D\psi(1)j$ and $D\psi(1)k$ are linearly independent.