Final answer:
To find the derivative of the given function at x = e, use the product rule and chain rule to differentiate y = e^{g(x)} / ln(x). Substituting the given values and simplifying, we find that y'(e) = 1/e, which is approximately 0.3679.
Step-by-step explanation:
The question asks for the derivative of the function y = e^{g(x)} / ln(x) evaluated at x = e, given that the differentiable function g satisfies g(e) = 0 and g'(e) = 2/e. To find y', we will use the product rule and the chain rule for differentiation
Firstly, let's differentiate y = e^{g(x)} / ln(x):
y' = (d/dx e^{g(x)}) / ln(x) + e^{g(x)} * -(1/x * 1/ln(x)^2)
Using the chain rule, this becomes:
y' = e^{g(x)}g'(x) / ln(x) - e^{g(x)} / (xln(x)^2).
Next, substituting x=e, g(e)=0, g'(e)=2/e into the differentiated equation, we get:
y' = e^0 * (2/e) / ln(e) - e^0 / (eln(e)^2)
= 2/e - 1/e = 1/e.
Therefore, y'(e) = 1/e, and since e ≈ 2.71828, the numerical answer is approximately y'(e) ≈ 0.3679.