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Derivatives Find and simplify the derivative of the following functions.$$f(x)=\frac{e^{x}}{e^{x}+1}$$

$\dfrac{e^{x}}{(e^{x}+1)^{2}}$

Calculus 1 / AB

Chapter 3

Derivatives

Section 4

The Product and Quotient Rules

Differentiation

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Find the derivative of the…

let's find the derivative of this function F of X. That is, we're going to be finding f prime of X so we can apply the quotient rule to this. Therefore, we first have the bottom term. That is, each the X plus one multiplied by the derivative of the top term just into the X minus the top term either the ex times derivative of the bottom term, which is just you. The X derivative of 10 and derivative of the X is you the X and all this is divided by the bottom term squared. That is, either the X plus one all squared. Remember those parentheses. They're very important. Now we have to simplify this. So first off, we can distribute through these parentheses, giving us E to the X squared, or either the two x plus ive the X minus another each the X squared or e to the two X, and this is divided by well, squaring this out, we would get e to the two X or you the X squared plus to eat the X plus one. Now, there's some more simplification to be done here. That is, as you can see, we can cancel on E to the two X and a negative each of the two x from the top, leaving us with just need to the X on the top and then on the bottom. We can't simplify any further, so we just have each the two x plus two e to the X plus one. Thus, since we can't simplify any further, this is our final answer.

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