Function $f\in L^1(\mathbb{R})$ which is absolutely continous but $f'\notin L^1(\mathbb{R})$

real-analysis lp-spaces distribution-theory absolute-continuity

Just an idea, an analytic function in $L^1(\mathbb{R})$ with derivative not in $L^1(\mathbb{R})$ would work. How about $$f(x) = \frac{\sin x^2}{x^2+1}\\ f'(x) = \frac{2 x \cos x^2}{x^2+1} - \frac{\sin x^2} {(x^2 + 1)^2}$$

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