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What is the derivative of #lnx / x#?

Answer 1

Cameron Alexander

Using quotient rule, which states that for #y=f(x)/g(x)#, #(dy)/(dx)=(f'(x)g(x)=f(x)g'(x))/g(x)^2#

Solving:

#(dy)/(dx)=((1/x)x-lnx(1))/x^2#

#(dy)/(dx)=(1-lnx)/x^2#

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Answer 2

Charles Arocha

The derivative of ( \frac{\ln(x)}{x} ) can be found using the quotient rule. The quotient rule states that if ( f(x) = \frac{g(x)}{h(x)} ), then ( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} ).

Let ( g(x) = \ln(x) ) and ( h(x) = x ).

Then ( g'(x) = \frac{1}{x} ) and ( h'(x) = 1 ).

Using the quotient rule, we have:

[ f'(x) = \frac{\frac{1}{x} \cdot x - \ln(x) \cdot 1}{(x)^2} ]

[ = \frac{1 - \ln(x)}{x^2} ]

So, the derivative of ( \frac{\ln(x)}{x} ) is ( \frac{1 - \ln(x)}{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.