How do you find the first and second derivative of #y=1/(1+e^-x)#?
There are 2 approaches to differentiating this function.
I'll use approach (1) you could perhaps try approach (2). The result will be the same.
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To find the first derivative of ( y = \frac{1}{1 + e^{-x}} ), we use the chain rule.
Let ( u = 1 + e^{-x} ), then ( y = u^{-1} ).
Thus, using the chain rule, the derivative of ( y ) with respect to ( x ) is:
[ \frac{dy}{dx} = \frac{d}{dx} \left( u^{-1} \right) = -u^{-2} \frac{du}{dx} ]
Now, differentiate ( u ) with respect to ( x ):
[ \frac{du}{dx} = \frac{d}{dx} (1 + e^{-x}) = 0 - e^{-x}(-1) = e^{-x} ]
Substitute ( u ) and ( \frac{du}{dx} ) back into the derivative:
[ \frac{dy}{dx} = -\frac{1}{(1 + e^{-x})^2} \cdot e^{-x} ]
To find the second derivative, differentiate ( \frac{dy}{dx} ) with respect to ( x ):
[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( -\frac{1}{(1 + e^{-x})^2} \cdot e^{-x} \right) ]
Using the product rule and chain rule, the derivative is:
[ \frac{d^2y}{dx^2} = -\frac{d}{dx} \left( \frac{1}{(1 + e^{-x})^2} \right) \cdot e^{-x} - \frac{1}{(1 + e^{-x})^2} \cdot \frac{d}{dx}(e^{-x}) ]
Now, differentiate each term:
[ \frac{d}{dx} \left( \frac{1}{(1 + e^{-x})^2} \right) = \frac{2}{(1 + e^{-x})^3} \cdot e^{-x} ]
[ \frac{d}{dx}(e^{-x}) = -e^{-x} ]
Substitute these derivatives back:
[ \frac{d^2y}{dx^2} = -\frac{2}{(1 + e^{-x})^3} \cdot e^{-2x} + \frac{e^{-x}}{(1 + e^{-x})^2} ]
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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