f'(x) of f(x)= 4e^x/1-e^x

(bottom derivative of top - top derivative of bottom)/bottom squared

4 [ (1-e^x)e^x - e^x (-e^x) ] / (1-e^x)^2

4 [ e^x -e^2x + e^2x ]/(1-e^x)^2

4 e^x /(1-e^x^2)
interesting, check carefully

4 e^x /(1-e^x^2)^2

To find the derivative of the function f(x) = 4e^x / (1 - e^x), we can use the quotient rule. The quotient rule states that if we have a function in the form of f(x) = g(x) / h(x), then its derivative f'(x) can be calculated as:

f'(x) = (h(x) * g'(x) - g(x) * h'(x)) / (h(x))^2

Let's apply this rule to find the derivative of f(x).

First, we need to find the derivative g'(x) and h'(x). In this case:

g(x) = 4e^x
h(x) = 1 - e^x

Differentiating g(x) with respect to x, we get:
g'(x) = 4e^x (since the derivative of e^x is e^x)

Differentiating h(x) with respect to x, we get:
h'(x) = 0 - (-e^x) = e^x (using the power rule and the fact that the derivative of a constant, in this case, 1, is 0)

Now, we have all the necessary values to substitute them into the quotient rule formula:

f'(x) = (h(x) * g'(x) - g(x) * h'(x)) / (h(x))^2
= ((1 - e^x) * (4e^x) - (4e^x) * (e^x)) / (1 - e^x)^2

Simplifying further:

f'(x) = (4e^x - 4e^2x - 4e^2x) / (1 - 2e^x + e^2x)
= (-8e^2x + 4e^x) / (1 - 2e^x + e^2x)

Therefore, the derivative of f(x) = 4e^x / (1 - e^x) is f'(x) = (-8e^2x + 4e^x) / (1 - 2e^x + e^2x).