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What is the first differential of #f(x) = 1/(x(1+lnx)# ?

Answer 1

Samantha Adkison

#-(2+lnx)/(x^2(1+lnx)^2#

#f(x) = 1/(x(1+lnx)#

#= (x*(1+lnx))^-1#

Applying the power rule and the chain rule:

#= -(x*(1+lnx))^-2 * d/dx(x*(1+lnx))#

#= -(x*(1+lnx))^-2 * x*(0+1/x)+(1+lnx)*1# [Product rule and standard differential]

#= -(x*(1+lnx))^-2 * (1 +1 +lnx)#

#= -(2+lnx)/(x^2(1+lnx)^2#

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Answer 2

Cameron Alexander

The first differential of ( f(x) = \frac{1}{x(1+\ln x)} ) is ( f'(x) = -\frac{1 + \ln x}{x^2(1 + \ln x)^2} ).

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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.