How do you find the first and second derivative of # {ln(x)}^2#?
This is our first derivative. We obtain the second derivative by differentiating the first derivative.
In summary:
Practice exercises:
Bonus:
Determine the third derivatives of the functions above, if they exist.
Good luck, and hopefully this helps!
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To find the first and second derivatives of ( (\ln(x))^2 ):

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

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