Dixie Blake

2023-03-16

How to find the derivative of $y=\frac{1}{1+{e}^{x}}$?

Joselyn Arias

Beginner2023-03-17Added 5 answers

$\frac{dy}{dx}=-{e}^{x}{(1+{e}^{x})}^{-2}$

Explanation:

Differentiate using the chain rule

Given $y=f\left(g\left(x\right)\right)$

$\frac{dy}{dx}=f\prime \left(g\left(x\right)\right)\cdot (g\prime \left(x\right))\leftarrow$ chain rule

$y={(1+{e}^{x})}^{-1}$

$\frac{dy}{dx}=(-{(1+{e}^{x})}^{-2})\cdot {e}^{x}$

$=-\frac{{e}^{x}}{{(1+{e}^{x})}^{2}}$

Explanation:

Differentiate using the chain rule

Given $y=f\left(g\left(x\right)\right)$

$\frac{dy}{dx}=f\prime \left(g\left(x\right)\right)\cdot (g\prime \left(x\right))\leftarrow$ chain rule

$y={(1+{e}^{x})}^{-1}$

$\frac{dy}{dx}=(-{(1+{e}^{x})}^{-2})\cdot {e}^{x}$

$=-\frac{{e}^{x}}{{(1+{e}^{x})}^{2}}$

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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