How do you find the first and second derivative of #ln(lnx^2)#?
Using the wellknown Rules of Log. Fun., we have,
By the Chain Rule, then,
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To find the first and second derivatives of ( \ln(\ln(x^2)) ):

First Derivative: ( \frac{d}{dx}(\ln(\ln(x^2))) = \frac{1}{\ln(x^2)} \cdot \frac{d}{dx}(\ln(x^2)) ) Apply chain rule and derivative of natural logarithm: ( = \frac{1}{\ln(x^2)} \cdot \frac{1}{x^2} \cdot 2x )

Second Derivative: ( \frac{d^2}{dx^2}(\ln(\ln(x^2))) = \frac{d}{dx} \left( \frac{1}{\ln(x^2)} \cdot \frac{1}{x^2} \cdot 2x \right) ) Apply quotient rule and derivative of ( \frac{1}{\ln(x^2)} ): ( = \frac{2}{x^3\ln(x^2)} + \frac{1}{x^4} \cdot 2x \cdot \left( \frac{1}{x^2} \right) \cdot \frac{d}{dx}(\ln(x^2)) ) ( = \frac{2}{x^3\ln(x^2)}  \frac{2}{x^3} )
Therefore, the first derivative is ( \frac{2}{x(\ln(x))^2} ) and the second derivative is ( \frac{2}{x^3(\ln(x))^2}  \frac{2}{x^3} ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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