Find the derivative of y = xe ^ -kx

Question

I know I need to use the chain rule but I never get the right answer when I try on this problem

Answer 1

Hello once again, Mia! 

 

 

Based on your previous questions, my guess is you understand the product and chain rules already. It would be my guess (and please feel free to message me and correct me if I'm wrong), you don't understand how to apply the chain rule specifically to e^-kx. Thus, I'll just explain that particular part of the problem because Michael has already done a wonderful job working through the rest of the problem.

 

JUST TO BE CLEAR, I'm only working through finding the derivative of e^-kx

 

 

 

A reiteration of the Chain Rule:

F(x) = f'(g(x)) * g'(x)

 

g = -kx

g' = -k * 1

 

You might be thinking, "Wait, why did that -k stick around? I thought variables just turned to constants?" This, I believe, is the heart of your confusion. -k is actually a CONSTANT. Let's pretend k = 3

 

h = -3x

h'     =      -3 * 1    =    -3

 

Turning back to the problem at hand, let's now apply the chain rule:

 

f'(e^-kx)   *    g'(-kx)

 

Well, we already know g' --- it's -k. The derivative of e^any variable or number has the property of always not changing from the original function. THUS,

 

e^-kx       *        -k

 

= -ke^-kx

 

 

Please let me know if you have any questions!

Answer 2

so if we apply the product rule 

 

u = x      v = e^-kx

 

u' = 1      v'= -ke^-kx

 

so putting it together we get 

 

e^-kx+x(-ke^-kx)

 

if we factor we get 

 

e^-kx(1-kx)