what is the second derivative of:

Y= 50/ (1 + 4e^ -.1t)

To find the second derivative of the function Y = 50 / (1 + 4e^(-0.1t)), we'll need to apply the chain rule twice.

Step 1: Find the first derivative
To find the first derivative, we can use the quotient rule. The quotient rule states that if we have a function f(t) = g(t) / h(t), then the derivative of f(t) is given by:
f'(t) = (g'(t) * h(t) - g(t) * h'(t)) / (h(t))^2

Using the quotient rule on our function Y = 50 / (1 + 4e^(-0.1t)), we get:
Y' = [(0 * (1 + 4e^(-0.1t))) - (50 * (-0.4e^(-0.1t)))] / (1 + 4e^(-0.1t))^2
Simplifying this, we have:
Y' = (200e^(-0.1t)) / (1 + 4e^(-0.1t))^2

Step 2: Find the second derivative
To find the second derivative, we differentiate the first derivative obtained in step 1. We can use the quotient rule again:
Y'' = [((200e^(-0.1t))(1 + 4e^(-0.1t))^2 - (200e^(-0.1t))(2(1 + 4e^(-0.1t))(0.4e^(-0.1t))))] / (1 + 4e^(-0.1t))^4
Simplifying this further yields the second derivative of the given function.

With these steps, we have found the second derivative of Y = 50 / (1 + 4e^(-0.1t)).