Question

G(x) = e^x/e^x-x
g'(x) =?

Answer 1

To find the derivative of g(x) = e^x / (e^x - x), we can first simplify the function using the quotient rule, and then apply the chain rule.

Using the quotient rule, we get:

g'(x) = [ (e^x)(e^x - x)' - (e^x - x)(e^x)' ] / (e^x - x)^2

g'(x) = [ (e^x)(-1) - (e^x - x)(e^x) ] / (e^x - x)^2 (using (e^x)' = e^x)

g'(x) = [ -e^x + xe^x ] / (e^x - x)^2

Now, applying the chain rule, we get:

g'(x) = [ (-e^x + xe^x)(e^x - x)' ] / (e^x - x)^2

g'(x) = [ (-e^x + xe^x)(e^x - 1) ] / (e^x - x)^2

Therefore:

g'(x) = [ (-e^x + xe^x)(e^x - 1) ] / (e^x - x)^2.