TheHamburgler1 said:Homework Statement
Find the derivative of (1+e^-x)^-1
Homework Equations
The Attempt at a Solution
I can't seem to get anywhere with this. Should I be looking for a property of something like the cosh function to apply to this?
Thanks
TheHamburgler1 said:Of course, silly me.
So I get (e^(-x))((1+e^-x)^-2)
Can this be simplified?
-Cheers
The derivative of (1+e^-x)^-1 is -(1+e^-x)^-2 * e^-x.
To find the derivative of (1+e^-x)^-1, you can use the power rule and chain rule. First, rewrite the function as (1+e^-x)^-1 = (1+e^-x)^-1 * 1. Then, use the power rule to find the derivative of the numerator, which is 0. Next, use the chain rule to find the derivative of the denominator, which is -(1+e^-x)^-2 * e^-x. Finally, combine the two derivatives to get the overall derivative of -(1+e^-x)^-2 * e^-x.
The graph of the derivative of (1+e^-x)^-1 is a negative exponential curve with a maximum at (0, 1) and an asymptote at x=-1. As x approaches infinity, the derivative approaches 0, and as x approaches -1, the derivative approaches negative infinity.
The derivative of (1+e^-x)^-1 can be used in various fields of science and engineering, such as physics, chemistry, and biology. It can be used to model the rate of change of a system over time, such as the decay of a radioactive substance or the growth of bacteria. It can also be used in optimization problems to find maximum or minimum values of a function.
The derivative of (1+e^-x)^-1 is closely related to the inverse function rule and the logarithmic rule. It can also be used in integration problems to find the original function. Additionally, it is a special case of the chain rule, which is a fundamental concept in calculus.