= Solution
The endpoint agreement $f(1)=f(0)=0$ makes the boundary terms vanish when integrating the <Fourier coefficients> of $f'$ by parts. If $a'_n,b'_n$ denote these coefficients, then
$$
a'_n=2\pi n\,b_n=0,\qquad
b'_n=-2\pi n\,a_n=\frac2{\pi n},\qquad a'_0=0.
$$
These are exactly the coefficients obtained by differentiating the preceding <Fourier series>:
$$
\boxed{G(x)=\frac2\pi\sum_{n=1}^\infty\frac{\sin(2\pi nx)}n.}
$$
On every compact subinterval of $(0,1)$ the differentiated series converges uniformly by the <Dirichlet test>, since partial sums of the sine factors are uniformly bounded there. The <Termwise differentiation of a Fourier series> is therefore valid away from integers and gives $1-2x$ on $(0,1)$. At an integer the periodic original has a corner, so the differentiated series is not asserting the existence of an ordinary derivative there.
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