How do you differentiate #g(x) = (1/x^3)*sqrt(x-xe^(x))# using the product rule?

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Answer 1 · 2016-08-08T12:19:47

# (1-e^x -xe^x)/(2x^3sqrt(x-xe^x)) -sqrt(x-xe^x)/(3x^4) #

First let me rewrite this
#(x^-3) * (x-xe^x)^(1/2)#

#(f(x) * g(x))prime = f(x)primeg(x) +g(x)primef(x)#

#d/dx (x^-3) * (x-xe^x)^(1/2) = #

#-3x^-4 * (x-xe^x)^(1/2) + x^-3 * (1/2) (x-xe^x)^(-1/2) d/dx (x-xe^x) #

now just focusing on the last part because we need to do the power rule again
#d/dx (x-xe^x) =1-e^x +xe^x #

now we can place this back in

#-3x^-4 * (x-xe^x)^(1/2) + x^-3 * (1/2) (x-xe^x)^(-1/2) (1-e^x -xe^x) #

now i rearrange it a bit

# (1-e^x -xe^x)/(2x^3sqrt(x-xe^x)) -sqrt(x-xe^x)/(3x^4) #