I want to prove: $|f'(a)| \leq \frac{M}{1-|a|^2}$ so, what is the technic for use this Hint in down:

Question

We have $f$ is a function holomorphic on $D(0,1)$, bounded by $M>0$, how to use this hint : $$\frac{1}{1-|a|^2}= \frac{1}{2i\pi} \int_{|z|=1} \frac{1}{z|z-a|^2} \,dz$$