What are the first and second derivatives of #f(x)=ln((x1)^2/(x+3))^(1/3) #?
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To find the derivatives of ( f(x) = \ln\left(\frac{(x1)^2}{x+3}\right)^{1/3} ), we first simplify the function using logarithmic properties and then differentiate term by term using the chain rule and the power rule for differentiation.

First, simplify the function: ( f(x) = \ln\left(\frac{(x1)^2}{x+3}\right)^{1/3} ) ( = \frac{1}{3} \ln\left(\frac{(x1)^2}{x+3}\right) ) ( = \frac{1}{3} \left( \ln((x1)^2)  \ln(x+3) \right) ) ( = \frac{1}{3} \left( 2\ln(x1)  \ln(x+3) \right) ) ( = \frac{2}{3} \ln(x1)  \frac{1}{3} \ln(x+3) )

Now, differentiate term by term: ( f'(x) = \frac{2}{3} \cdot \frac{1}{x1}  \frac{1}{3} \cdot \frac{1}{x+3} ) ( = \frac{2}{3(x1)}  \frac{1}{3(x+3)} ) ( = \frac{2(x+3)  (x1)}{3(x1)(x+3)} ) ( = \frac{2x + 6  x + 1}{3(x1)(x+3)} ) ( = \frac{x + 7}{3(x1)(x+3)} )

For the second derivative, we differentiate ( f'(x) ): ( f''(x) = \frac{d}{dx} \left( \frac{x + 7}{3(x1)(x+3)} \right) ) ( = \frac{3(x1)(x+3)  (x+7)(3x+3)}{3(x1)^2(x+3)^2} ) ( = \frac{3(x^2 + 2x  x  3)  (3x^2 + 3x + 21x + 21)}{3(x1)^2(x+3)^2} ) ( = \frac{3(x^2 + x  3)  (3x^2 + 24x + 21)}{3(x1)^2(x+3)^2} ) ( = \frac{3x^2 + 3x  9  3x^2  24x  21}{3(x1)^2(x+3)^2} ) ( = \frac{21x  30}{3(x1)^2(x+3)^2} ) ( = \frac{7(x + 10)}{(x1)^2(x+3)^2} )
So, the first derivative is ( f'(x) = \frac{x + 7}{3(x1)(x+3)} ) and the second derivative is ( f''(x) = \frac{7(x + 10)}{(x1)^2(x+3)^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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