Find 1st and 2nd Derivative.

Affinity · November 21st 2006, 06:48 PM

Find the first derivative and then the second derivative of

Thank you in advance!

**ThePerfectHacker** · November 21st 2006, 07:19 PM

Originally Posted by Affinity

Find the first derivative and then the second derivative of

Thank you in advance!

$y=\frac{e^x}{x^2}$

$y'=\frac{(e^x)'(x^2)-(e^x)(x^2)'}{(x^2)^2}$
$y'=\frac{x^2e^x-2xe^x}{x^4}=\frac{xe^x(x-2)}{x^4}=\frac{e^x(x-2)}{x^3}$

$y''=\frac{[e^x(x-2)]'(x^3)-[e^x(x-2)](x^3)'}{(x^3)^2}$
$y''=\frac{[(e^x)'(x-2)+e^x(x-2)']x^3-e^x(x-2)(x^3)'}{x^6}$
$y''=\frac{[e^x(x-2)+e^x]x^3-3x^2e^x(x-2)}{x^6}$
$y''=\frac{e^x(x-2)x^3+x^3e^x-3x^2e^x(x-2)}{x^6}=\frac{e^x[(x-2)x^3+x^3-3x^2(x-2)]}{x^6}$
$y''=\frac{e^x(x^4-2x^3+x^3-3x^3+6x^2)}{x^6}=\frac{e^x(x^4-4x^3+6x^2)}{x^6}=$ $\frac{x^2e^x(x^2-4x+6)}{x^6}$
$y''=\frac{e^x(x^2-4x+6)}{x^4}$

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