I'm going over the proof of the spectral theorem for compact symmetric operators in Hilbert space in Lax. Let $A$ be a compact symmetric operator on a Hilbert space to itself. Define the Rayleigh quotient to be
$R_A(x) = \frac{(Ax, x)}{\|x\|^2}$
Let $z$ be vector that maximizes the quadratic form $(Ax, x)$ over the unit ball. Let $w$ be arbitrary. Now the text claims that $R(z + tw)$ is differentiable, and since it achieves its maximum at $t=0$, it's $t$-derivative is zero, and we have
$\frac{(Aw, z) + (Az, w)}{\|z\|^2} - (Az, z) \frac{(w, z) + (z, w)}{\|z\|^4} = 0$
I don't see why the given function is differentiable, nor the computation of its derivative.