Quotient rule
Proof by logarithmic differentiation

Let h (x) = f (x) g (x). {\displaystyle h(x)={\frac {f(x)}{g(x)}}.} Taking the absolute value and natural logarithm of both sides of the equation gives

Applying properties of the absolute value and logarithms,

Taking the logarithmic derivative of both sides,

Solving for h ′ (x) {\displaystyle h'(x)} and substituting back f (x) g (x) {\displaystyle {\tfrac {f(x)}{g(x)}}} for h (x) {\displaystyle h(x)} gives:

Taking the absolute value of the functions is necessary for the logarithmic differentiation of functions that may have negative values, as logarithms are only real-valued for positive arguments. This works because d d x (ln | u |) = u ′ u {\displaystyle {\tfrac {d}{dx}}(\ln |u|)={\tfrac {u'}{u}}}, which justifies taking the absolute value of the functions for logarithmic differentiation.
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