# What is the second derivative of #f(x)=x/(x^2+1)#?

graph{(2x^5-4x^3-6x)/(x^2+1)^4 [-10, 10, -5, 5]}

graph{x/(x^2+1) [-10, 10, -5, 5]}

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To find the second derivative of ( f(x) = \frac{x}{x^2 + 1} ), first find the first derivative using the quotient rule:

( f'(x) = \frac{(x^2 + 1) - x(2x)}{(x^2 + 1)^2} )

Simplify this to get:

( f'(x) = \frac{1 - x^2}{(x^2 + 1)^2} )

Now, differentiate ( f'(x) ) with respect to ( x ) again to find the second derivative:

( f''(x) = \frac{(1 - x^2)'(x^2 + 1)^2 - (1 - x^2)(2(x^2 + 1)(2x))}{(x^2 + 1)^4} )

Simplify this expression to get the second derivative.

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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.

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.

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.

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