Logarithmic Differentiation - Applications - Quotients

Quotients

A natural logarithm is applied to a quotient of two functions

to transform the division into a subtraction

Differentiate by applying the chain and the sum rules

and, after rearranging, get

f'(x) = f(x)\times \Bigg\{\frac{g'(x)}{g(x)}-\frac{h'(x)}{h(x)}\Bigg\}=
\frac{g(x)}{h(x)}\times \Bigg\{\frac{g'(x)}{g(x)}-\frac{h'(x)}{h(x)}\Bigg\}

After multiplying out and using the common denominator formula the result is the same as if after applying the quotient rule directly to .

Read more about this topic:  Logarithmic Differentiation, Applications

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