How do you find the second derivative of $f(x) = e^x/x^2$ ?

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Answer 1 · 2016-09-28T07:20:19

$(d^2f)/(dx^2)=(e^x(x^2-4x+6))/x^4$

Let us first find the first derivative of $f(x)=e^x/x^2$ using quotient rule

that if $f(x)=(g(x))/(h(x))$

then $(df)/(dx)=((dg)/(dx)xxh(x)-(dh)/(dx)xxg(x))/(h(x))^2$ .

$(df)/(dx)=(x^2xxe^x-e^x xx2x)/(x^2)^2=(x^2e^x-2xe^x)/x^4=(xe^x(x-2))/x^4=(e^x(x-2))/x^3$

and second derivative $(d^2f)/(dx^2)=d/(dx)((df)/(dx))=d/(dx)(e^x(x-2))/x^3$

= $(x^3xx(e^x(x-2)+e^x xx1)-e^x(x-2)xx3x^2)/x^6$

= $(e^x x^3(x-2)+e^x x^3-3x^2e^x(x-2))/x^6$

= $(e^x x^2(x(x-2)+x-3(x-2)))/x^6$

= $(e^x(x^2-2x+x-3x+6))/x^4$

= $(e^x(x^2-4x+6))/x^4$

How do you find the second derivative of f(x) = e^x/x^2f(x) = e^x/x^2?