Question

How do you differentiate `f(x)=(x^2)/(e^(x^-3))` using the quotient rule?

Answer 1

`f'(x) = ( x(2-x) ) / (e^(x-3))`

Explanation:

The quotient rule is `d/dx(u/v)=(v(du)/dx - u(dv)/dx)/v^2`

So, `f(x)=x^2/e^(x-3)`
`:. f'(x) = ( e^(x-3) d/dx(x^2)-x^2d/dx(e^(x-3)) ) / (e^(x-3))^2`
`:. f'(x) = ( e^(x-3) (2x)-x^2(e^(x-3)) ) / (e^(x-3))^2`
`:. f'(x) = ( e^(x-3) (2x-x^2) ) / (e^(x-3))^2`
`:. f'(x) = ( (2x-x^2) ) / (e^(x-3))`
`:. f'(x) = ( x(2-x) ) / (e^(x-3))`