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What is the derivative of #y=ln(1/x)#?

Answer 1

Charles Anctil

#y'=-1/x#

Full solution

#y=ln(1/x)#

This can be solved in two different ways,

Explanation (I)

The simplest one is, using logarithm identity,

#log(1/x^y)=log(x^-y)=-ylog (x)#, similarly following for problem,

#y=-ln(x)#

#y'=-(ln(x))'#

#y'=-1/x#

Explanation (II)

Using Chain Rule,

#y'=(ln(1/x))'#

#y'=1/(1/x)*(-1/x^2)=-1/x#

#y'=-1/x#

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Answer 2

Ezra Adams

The derivative of ( y = \ln(1/x) ) with respect to ( x ) is:

[ \frac{dy}{dx} = -\frac{1}{x} ]

This can be obtained using the chain rule and the derivative of the natural logarithm function.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.