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Exponential and Log, Logarithmic Differentiation

الكلية كلية الهندسة     القسم  الهندسة الميكانيكية     المرحلة 1
أستاذ المادة احمد كاظم حسين الحميري       13/07/2018 15:01:57
Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does not take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have

{\displaystyle (\log uv) =(\log u+\log v) =(\log u) +(\log v) .\!} (\log uv) =(\log u+\log v) =(\log u) +(\log v) .\!
So for positive-real-valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors. But we can also use the Leibniz law for the derivative of a product to get

{\displaystyle {\frac {(uv) }{uv}}={\frac {u v+uv }{uv}}={\frac {u }{u}}+{\frac {v }{v}}.\!} {\frac {(uv) }{uv}}={\frac {u v+uv }{uv}}={\frac {u }{u}}+{\frac {v }{v}}.\!
Thus, it is true for any function that the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors (when they are defined).

A corollary to this is that the logarithmic derivative of the reciprocal of a function is the negation of the logarithmic derivative of the function:

{\displaystyle {\frac {(1/u) }{1/u}}={\frac {-u /u^{2}}{1/u}}=-{\frac {u }{u}},\!} {\frac {(1/u) }{1/u}}={\frac {-u /u^{2}}{1/u}}=-{\frac {u }{u}},\!
just as the logarithm of the reciprocal of a positive real number is the negation of the logarithm of the number.

More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives of the dividend and the divisor:

{\displaystyle {\frac {(u/v) }{u/v}}={\frac {(u v-uv )/v^{2}}{u/v}}={\frac {u }{u}}-{\frac {v }{v}},\!} {\frac {(u/v) }{u/v}}={\frac {(u v-uv )/v^{2}}{u/v}}={\frac {u }{u}}-{\frac {v }{v}},\!
just as the logarithm of a quotient is the difference of the logarithms of the dividend and the divisor.

Generalising in another direction, the logarithmic derivative of a power (with constant real exponent) is the product of the exponent and the logarithmic derivative of the base:

{\displaystyle {\frac {(u^{k}) }{u^{k}}}={\frac {ku^{k-1}u }{u^{k}}}=k{\frac {u }{u}},\!} {\frac {(u^{k}) }{u^{k}}}={\frac {ku^{k-1}u }{u^{k}}}=k{\frac {u }{u}},\!
just as the logarithm of a power is the product of the exponent and the logarithm of the base.

In summary, both derivatives and logarithms have a product rule, a reciprocal rule, a quotient rule, and a power rule (compare the list of logarithmic identities); each pair of rules is related through the logarithmic derivative.

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