How do you find the first and second derivative of #ln(x/(x^2+1))#?
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To find the first and second derivatives of ln(x/(x^2+1)), we use the quotient rule for differentiation:

First Derivative: f'(x) = [1/(x/(x^2+1))][(x/(x^2+1))'  (x^2+1)'(x/(x^2+1))^2] = [(x^2+1)  x(2x)] / [(x^2+1)^2] = (x^2 + 1  2x^2) / (x^2 + 1)^2 = (1  x^2) / (x^2 + 1)^2

Second Derivative: To find the second derivative, differentiate the first derivative with respect to x: f''(x) = [(2x)(x^2 + 1)^2  (1  x^2)(2(x^2 + 1)(2x))] / (x^2 + 1)^4 = (2x(x^2 + 1)^2  2(1  x^2)(2x^3 + 2x)) / (x^2 + 1)^4 = (2x(x^2 + 1)^2  4x(1  x^2)(x^2 + 1)) / (x^2 + 1)^4 = (2x(x^4 + 2x^2 + 1)  4x(x^2 + 1  x^4  1)) / (x^2 + 1)^4 = (2x^5 + 4x^3 + 2x  4x^3  4x) / (x^2 + 1)^4 = (2x^5  4x^4  2x) / (x^2 + 1)^4
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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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