Let f(x) = (lne^2x)/(x-1) for x>1. If g(x) is the inverse of f(x0, then g'(3) = ?
A. 2
B. 1
C. o
D. -1
or E. -2
if you can show/explain how to get that please, thanks
To find the value of g'(3), we can follow these steps:
1. Start by finding the inverse function g(x) of f(x).
2. Once we have g(x), we can differentiate it to find g'(x).
3. Finally, substitute x = 3 into g'(x) to obtain the value of g'(3).
Let's now go through these steps in detail:
1. Inverse Function:
To find the inverse function g(x) of f(x), we need to interchange x and y in the equation and solve for y.
Given: f(x) = (ln(e^2x))/(x-1)
Let y = f(x) and interchange x and y: x = (ln(e^2y))/(y-1)
Now, solve for y:
x(y-1) = ln(e^2y)
xy - x = 2yln(e)
xy - x = 2y (Because ln(e) = 1)
xy - 2y = x
y(x - 2) = x
y = x / (x - 2)
Therefore, the inverse function g(x) of f(x) is:
g(x) = x / (x - 2)
2. Differentiation of g(x):
Next, we differentiate the function g(x) to find g'(x).
To differentiate g(x), we can use the quotient rule, which states:
If h(x) = u(x) / v(x), then h'(x) = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2
Let's apply this rule to g(x) = x / (x - 2):
Taking u(x) = x and v(x) = (x - 2)
First, find u'(x) and v'(x):
u'(x) = 1 (derivative of x with respect to x)
v'(x) = 1 (derivative of (x - 2) with respect to x)
Now, apply the quotient rule:
g'(x) = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2
= ((1)(x - 2) - (x)(1)) / (x - 2)^2
= (x - 2 - x) / (x - 2)^2
= -2 / (x - 2)^2
3. Substitute x = 3:
Finally, substitute x = 3 into the expression for g'(x) to find g'(3):
g'(3) = -2 / (3 - 2)^2
= -2 / 1^2
= -2 / 1
= -2
Therefore, g'(3) = -2.
Hence, the correct option is E. -2.