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**eddieg0304** Ok so I was wondering if anyone could help me out and check my work? i need to find the first and second derivatives of f(x) and I believe I am right but a kid that is in my class with me are having trouble checking our answer. Our teacher taught us how to do it by checking the graphs but we aren't sure that it's working out properly. Any help would be greatly appreciated.

$\displaystyle f(x)= 2x /{x}^{2} + 1 $

$\displaystyle f`(x)= ({x}^{2} + 1)(2)-(2x)(2x)/({{x}^{2} + 1})^{2 } $ found derivative

$\displaystyle f`(x)= ({2x}^{2} + 2 - {4x}^{2 }) /({{x}^{2} + 1})^{2 }$then simplified it

$\displaystyle f`(x)= ({-2x}^{2}+2)/ ({{x}^{2} + 1})^{2 } $

that is what I got for my first derivative

and this is my second:

$\displaystyle f``(x)= ({{x}^{2} + 1})^{2 }(-4x)-({-2x}^{2} + 2)(2({x}^{2} + 1})(2x)/ {({{x}^{2} + 1})^{2 }}^{ 2} $ second derivative

$\displaystyle f``(x)=({-4x}^{ 5}-{8x}^{3 }-4x)+({8x}^{5 }+{8x}^{ 3}-{8x}^{3 }- 8x)/( {{x}^{ 2}+1 })^{4 } $ simplified

$\displaystyle f``(x)=({4x}^{5 }- {8x}^{ 3} - 4x)/ ( {{x}^{ 2}+1 })^{4 } $ simplifying some more

$\displaystyle f``(x)= (4x({x}^{ 4}-{2x}^{ 2}-1))/ ( {{x}^{ 2}+1 })^{4 } $

and this is what I got for my second derivative

I was wondering if anyone could help me and tell me where I went wrong if I went wrong, or if this is completely correct. Also was wondering if anyone knew how the graphs of the first and second derivative compare to the original equation and if you could explain to me or point me in the right direction for info on that.