differentation of expotential functions

Question

Find the derivative of the function.
f(x) = x/e^x

Answer 1

You can either find this derivative using the product or quotient rules. I'm personally terrible at remembering the order of the quotient rule, so I usually prefer using the product rule whenever possible (and it's nearly always possible. The algebra is sometimes harder though.)

 

To find this derivative using the product rule, remember that you can rewrite 1/e^x as e^-x. Then your function is

f(x) = x*e^-x

 

The product rules states that the derivative of a product will be a number of terms equal to the factors of the original problem, with each term having the derivative of one factor.

e.g. if f(x) = a*b*c, then f'(x) = a'*b*c + a*b'*c + a*b*c'

Here, your factors are x and e^-x. When finding the derivatives, don't forget the chain rule on e^-x!

 

f'(x) = x'*e^-x + x* (e^-x)'

 = 1*e^-x + x*(-1*e^-x)

 = e^-x - x e^-x

 = e^-x(1-x)

 = (1-x)/e^x

 

Answer 2

You can rewrite the function as f(x) = xe^-x.

 

 

Use the product rule to get

 

dx/dx e^-x + x de^-x/dx = (1)e^-x + x(-e^-x) = e^-x - xe^-x = e^-x(1 - x)