Clarification of the proof of Corollary V.4.5 in Bhatia "Matrix Analysis"

The proof of the corollary is just noting that the given non-constant operator monotone function on $(-1, 1)$ satisfies $f'(0) \neq 0$. But the book just states that this is true because $f$ is monotone. This reason alone is insufficient.

To fix the proof, we apply Theorem V.3.4 to the matrix $A = diag(t, 0)$ with $t \neq 0$. Then the following matrix is positive semi-definite.
$$ f^{[1]}(A) = \begin{pmatrix} f'(t) & \frac{f(t) - f(0)}{t} \\ \frac{f(t) - f(0)}{t} & f'(0) \end{pmatrix} $$
As $f$ is non-constant, there is some $t \neq 0$ such that $f(t) \neq 0$. By positive semi-definiteness, $f'(t) \geq 0$ and $\det(f^{[1]}(A)) \geq 0$. Hence
$$ f'(t)f'(0) - \left(\frac{f(t) - f(0)}{t}\right)^2 \geq 0 $$
$$ f'(t)f'(0) \geq \left(\frac{f(t) - f(0)}{t}\right)^2 > 0 $$
This implies that $f'(0) > 0$.