Differentiate y = x/e^x

Question

Differentiate y = x/e^x

Answer 1

For this problem we need to use the quotient rule

 

u = x      u' = 1

 

v = e^x  v' = e^x

 

so then we can put it together 

 

dy/dx = vu'-uv'/v^2

 

dy/dx = e^x- x e^x/ (e^x)^2

 

you can factor the top 

 

dy/dx = e^x(1-x)/(e^x)^2

 

then we can simplify 

 

dy/dx = 1-x/e^x

Answer 2

You'll start off using the quotient rule which states:

 

dy/dx  (f/g)   =    f' * g  - g' * f

                                  g^2

 

 

The next step is to go through and fill out this formula:

 

f = x

f'(x) = 1

 

g = e^x

g'(e^x) = e^x            (no, this isn't a typo, the derivative of e^x is e^x)

 

THUS,

(1)

(1 * e^x) - (e^x * x)

          e^2x

 

 

(2)

e^x - xe^x
     e^2x

 

 

(3) Now we can factor to further simplify this:

e^x (1 - x)

    e^2x

 

 

(4)

1 - x

 e^x

 

Let me know if you have any questions!