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How do you differentiate #y=ln(x^2)#?

Answer 1

The derivative is #dy/dx=2/x#

As with some problems involving differentiation of logarithmic functions, this problem can be simplified using the laws of logarithms.

#y=ln(x^2)# #y=2lnx# #dy/dx=2*1/x=2/x#

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Answer 2

Adam Adkins

To differentiate ( y = \ln(x^2) ), apply the chain rule:

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

Let ( u = x^2 ), then ( \frac{du}{dx} = 2x ). Now differentiate ( \ln(u) ) with respect to ( u ), which gives ( \frac{1}{u} ), and substitute back ( u = x^2 ) to get ( \frac{1}{x^2} ).

Finally, multiply by ( \frac{du}{dx} ), which is ( 2x ), to get:

[ \frac{dy}{dx} = \frac{1}{x^2} \cdot 2x = \frac{2}{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.