What is the derivative of #f(x)=log(x)/x# ?

Answer 1

The derivative is #f'(x)=(1-logx)/x^2#.

This is an example of the the Quotient Rule:

Quotient Rule .

The quotient rule states that the derivative of a function #f(x)=(u(x))/(v(x))# is:

#f'(x)=(v(x)u'(x)-u(x)v'(x))/(v(x))^2#.

To put it more concisely:

#f'(x)=(vu'-uv')/v^2#, where #u# and #v# are functions (specifically, the numerator and denominator of the original function #f(x)#).

For this specific example, we would let #u=logx# and #v=x#. Therefore #u'=1/x# and #v'=1#.

Substituting these results into the quotient rule, we find:

#f'(x)=(x xx 1/x-logx xx 1)/x^2#

#f'(x)=(1-logx)/x^2#.

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Answer 2

To find the derivative of ( f(x) = \frac{\log(x)}{x} ), you can use the quotient rule, which 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} ). Applying this rule to ( f(x) = \frac{\log(x)}{x} ), where ( g(x) = \log(x) ) and ( h(x) = x ), we get:

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

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.

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.

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.

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