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How do you find the derivative of #y=(lnx)^2#?

Answer 1

Isabella Ackerman

#dy/dx=(2lnx)/x#

#y=(lnx)^2#

Use the Chain Rule:

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

#dy/dx=(2lnx)/x#

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Answer 2

Paisley Johnson

To find the derivative of ( y = (\ln(x))^2 ), you can use the chain rule along with the power rule.

First, differentiate the outer function ((\ln(x))^2) with respect to the inner function (\ln(x)). The derivative of (u^2) with respect to (u) is (2u). So, differentiate ((\ln(x))^2) with respect to (\ln(x)), which yields (2\ln(x)).

Now, differentiate the inner function (\ln(x)) with respect to (x), which gives (\frac{1}{x}).

Now apply the chain rule, multiplying these two derivatives together:

[ \frac{d}{dx} \left((\ln(x))^2\right) = 2\ln(x) \cdot \frac{1}{x} = \frac{2\ln(x)}{x} ]

So, the derivative of ( y = (\ln(x))^2 ) with respect to (x) 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.