# Thread: Find 1st and 2nd Derivative.

1. ## Find 1st and 2nd Derivative.

Find the first derivative and then the second derivative of
Thank you in advance!

2. Originally Posted by Affinity
Find the first derivative and then the second derivative of
Thank you in advance!
$y=\frac{e^x}{x^2}$

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

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