What is the derivative of #(ln x)^2#?

Answer 1

Emma Aldridge

Here I would use the Chain Rule deriving first the #()^2# and then the #ln#: #y'=2ln(x)*1/x=2/xln(x)#

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Answer 2

Charles Arocha

To find the derivative of ( (\ln(x))^2 ), you can use the chain rule.

Let ( u = \ln(x) ), then ( y = u^2 ).

Now, differentiate ( y = u^2 ) with respect to ( u ), then apply the chain rule:

[ \frac{dy}{du} = 2u ]

Now differentiate ( u = \ln(x) ) with respect to ( x ):

[ \frac{du}{dx} = \frac{1}{x} ]

Apply the chain rule:

[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 2u \cdot \frac{1}{x} ]

Substitute ( u = \ln(x) ) back in:

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

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

So, the derivative of ( (\ln(x))^2 ) is ( \frac{2\ln(x)}{x} ).

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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.