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# (a) If $f(x) = e^x/ (2x^2 + x + 1),$ find $f' (x).$(b) Check to see that your answers to part (a) are reasonable by comparing the graphs of $f, f',$ and $f".$

## (a) $f^{\prime}(x)=\frac{4 x}{\left(x^{2}+1\right)^{2}}$ $f^{\prime \prime}(x)=\frac{4\left(1-3 x^{2}\right)}{\left(x^{2}+1\right)^{3}}$(b) See solution

Derivatives

Differentiation

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### Video Transcript

he It's clears the when you read here. So you have up of X is equal to X square minus one over X square plus one. We're gonna use a quotient role. To find our derivative, we get X square plus one D over deep rex of X square minus one minus X square minus one D over DX of X square plus one all over X square plus one square has becomes equal to X square plus one arms to X minus X square minus one times two x over X square plus one square. This becomes equal to four X over X squared, plus one square. Now we're gonna apply the same rule to find the second derivative when we get X Square plus one square D over D X for X minus for X times D over DX for X square plus one square all over X square plus one square and you score that again and then this simplifies into four minus 12 X square over X square plus one cubed. Here you apply the chain rule. Now we're going to draw some graphs for part B. We're going to start with R F of X so we'll draw down in black, an uncle like this, and our red is gonna be our derivative first derivative, and then our second derivative is gonna be in green.

Derivatives

Differentiation

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