# How do you find the second derivative of #y=ln(x^2)# ?

Deriving two times.

The first derivative is

then we derive again

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To find the second derivative of ( y = \ln(x^2) ), first, find the first derivative using the chain rule:

[ \frac{d}{dx}[\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx} ]

where ( u = x^2 ).

So, the first derivative, ( y' ), is:

[ y' = \frac{1}{x^2} \cdot 2x = \frac{2}{x} ]

Now, differentiate ( y' ) with respect to ( x ) to find the second derivative, ( y'' ):

[ y'' = \frac{d}{dx}\left(\frac{2}{x}\right) = -\frac{2}{x^2} ]

Therefore, the second derivative of ( y = \ln(x^2) ) is:

[ y'' = -\frac{2}{x^2} ]

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