# Thread: Find 1st and 2nd Derivative.

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

Find the first derivative and then the second derivative of

2. Originally Posted by Affinity
Find the first derivative and then the second derivative of
$\displaystyle y=\frac{e^x}{x^2}$
$\displaystyle y'=\frac{(e^x)'(x^2)-(e^x)(x^2)'}{(x^2)^2}$
$\displaystyle y'=\frac{x^2e^x-2xe^x}{x^4}=\frac{xe^x(x-2)}{x^4}=\frac{e^x(x-2)}{x^3}$
$\displaystyle y''=\frac{[e^x(x-2)]'(x^3)-[e^x(x-2)](x^3)'}{(x^3)^2}$
$\displaystyle y''=\frac{[(e^x)'(x-2)+e^x(x-2)']x^3-e^x(x-2)(x^3)'}{x^6}$
$\displaystyle y''=\frac{[e^x(x-2)+e^x]x^3-3x^2e^x(x-2)}{x^6}$
$\displaystyle y''=\frac{e^x(x-2)x^3+x^3e^x-3x^2e^x(x-2)}{x^6}=\frac{e^x[(x-2)x^3+x^3-3x^2(x-2)]}{x^6}$
$\displaystyle 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}=$$\displaystyle \frac{x^2e^x(x^2-4x+6)}{x^6}$
$\displaystyle y''=\frac{e^x(x^2-4x+6)}{x^4}$