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Numerade Educator

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Problem 29 Medium Difficulty

Find $ f'(x) $ and $ f"(x). $

$ f(x) = \frac {x^2}{1 + e^x} $

Answer

$\frac{e^{2 x} x^{2} \cdot e^{x} x^{2}-4 e^{2 x} x-4 e^{x} x+4 e^{x}+2 e^{2 x}+2}{\left(1+e^{x}\right)^{3}}$

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Video Transcript

he It's clear. So when you raid here, So we're gonna take the derivative. Do you ever d x for X square over one plus eat Vex. You were gonna ply the quotient rule, which is equal to D over DX X square one plus B to the X minus d over dx one plus b to the x Times X square all over one plus e to the x square We simplify and we get to x times one plus eat the x minus Eat the X Times X square all over one plus eat the X square for the second derivative. We're also gonna ply the quotient rule Let me get tea over d x for two x times one plus eats the X minus eats the x x square comes one plus Eat the X Square dynasty over d x one plus e to the x square times two x comes one plus heat the x minus Eat the X Times X square This is all over one plus. Eat the X square and square that again, and when we simplify we get is this equal to eat? Two X times X square minus Eat the X X Square minus four e to the two x times X minus four e to the x Times acts plus four e to the X Plus two e two the two x plus two all over one plus eat the X cubed.