First, we'll determine the function
g(x):
g(x) = (x^2)*(e^x)*(x-1)/x^2 +
1
We'll simplify and we'll
get:
g(x) = (e^x)*(x-1) +
1
We'll differentiate the function g(x), with respect to
x:
g'(x) = [(e^x)*(x-1) +
1]'
We'll apply the product
rule:
g'(x) = (e^x)'*(x-1) +
(e^x)*(x-1)'
g'(x) = (e^x)*(x-1) +
(e^x)
We'll factorize by
e^x:
g'(x) =
(e^x)*(x-1+1)
We'll eliminate like terms inside
brackets:
g'(x) =
x*(e^x)
The first derivative of the function
g(x) is g'(x) = x*(e^x).
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