Answer:
To find g'(e), we need to find the derivative of g(x) with respect to x and then evaluate it at x = e.
Let's start by finding the derivative of g(x). g(x) can be written as (7 In x)/ (3x+4).
To find the derivative, we can use the quotient rule, which states that if we have a function of the form f(x) / g(x), then the derivative is given by:
f'(x)g(x) = f(x)g'(x)
(g(x))^2
Applying this rule to g(x) = (7 In x) / (3x+4), we get:
(7/x)*(3x+4)-(7 In x) *3
(3x+4)^2
Explanation:
Simplifying this expression, we get:
(21x + 28 - 21 In x)/(x(3x+4))^2
Now, let's evaluate this expression at x = e:
(21e + 28 - 21 In e) / (e(3e+4))^2
Since In e is equal to 1, we can simplify
further:
(21e + 28-21)/(e(3e+4))^2
(21e + 7) / (e(3e+4))^2
So, g'(e) is equal to (21e + 7) / (e(3e+4))}^2.
Please let me know if I can help you with anything else.