Derivation of the cross-entropy equation in Michael Nielsen's book
The author has taken activation function as sigmoid in this case. The derivative of this function can be re-written like

$$\sigma(z) = \frac{1}{1+e^{-x}}$$

whose derivative $$\sigma'(z) = \frac{e^{-x}}{(1+e^{-x})^2}$$ which can rewritten as $$\frac{1+e^{-x}-1}{(1+e^{-x})^2} \rightarrow \sigma(z) * (1 - \sigma(z))$$. This is subtly mentioned in eq.3 of chap1.