# How do you differentiate #y=sqrt(2-e^x)#?

Use the chain rule:

and

Substituting this into the chain rule:

Reverse the substitution for g:

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To differentiate ( y = \sqrt{2 - e^x} ), you can use the chain rule. The chain rule states that if you have a function inside another function, you differentiate the outer function first, then multiply it by the derivative of the inner function.

So, for ( y = \sqrt{2 - e^x} ), the outer function is the square root, and the inner function is ( 2 - e^x ).

The derivative of the outer function ( \sqrt{u} ) with respect to ( u ) is ( \frac{1}{2\sqrt{u}} ), and the derivative of the inner function ( u = 2 - e^x ) with respect to ( x ) is ( -e^x ).

Applying the chain rule, we get:

[ \frac{dy}{dx} = \frac{1}{2\sqrt{2 - e^x}} \cdot (-e^x) ]

So, the derivative of ( y = \sqrt{2 - e^x} ) with respect to ( x ) is:

[ \frac{dy}{dx} = \frac{-e^x}{2\sqrt{2 - e^x}} ]

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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.

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