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Derivatives of compositions

The following two formulae are special cases of the chain rule:

d d x f (| x |) = x | x | (f ′ (| x |)) {\displaystyle {d \over dx}f(|x|)={x \over |x|}(f'(|x|))}

if the absolute value is inside a function, and

d d x | f (x) | = f (x) | f (x) | f ′ (x) {\displaystyle {d \over dx}|f(x)|={f(x) \over |f(x)|}f'(x)}

if another function is inside the absolute value. In the first case, the derivative is always discontinuous at x = 0 {\textstyle x=0} in the first case and where f (x) = 0 {\textstyle f(x)=0} in the second case.