Verify that #f(x)# is differentiable at #x = 0#? #f(x) = x((e^(1//x)  1)/(e^(1//x) + 1))#
graph{x((e^(1/x)  1)/(e^(1/x) + 1)) [1.065, 1.0684, 0.533, 0.5335]}
The derivative is:
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To verify that ( f(x) ) is differentiable at ( x = 0 ), we need to check if the derivative exists at that point.
Given ( f(x) = x\frac{e^{\frac{1}{x}}  1}{e^{\frac{1}{x}} + 1} ):
 First, we check the continuity of ( f(x) ) at ( x = 0 ).
 Then, we find the derivative of ( f(x) ) with respect to ( x ).
 Finally, we evaluate the limit of the derivative as ( x ) approaches 0.
Let's proceed with the calculations:

( f(x) ) is a rational function, and both the numerator and denominator approach 0 as ( x ) approaches 0. Therefore, ( f(x) ) is continuous at ( x = 0 ).

To find the derivative of ( f(x) ), we use the quotient rule: [ f'(x) = \frac{(e^{\frac{1}{x}} + 1)(1)  (e^{\frac{1}{x}}  1)(1)}{(e^{\frac{1}{x}} + 1)^2} ] [ = \frac{2}{(e^{\frac{1}{x}} + 1)^2} ]

To evaluate the limit of ( f'(x) ) as ( x ) approaches 0, we substitute ( x = 0 ) into ( f'(x) ): [ \lim_{x \to 0} f'(x) = \frac{2}{(e^{\frac{1}{0}} + 1)^2} = \frac{2}{(e^{\infty} + 1)^2} = \frac{2}{(\infty + 1)^2} = \frac{2}{\infty} = 0 ]
Since the limit of ( f'(x) ) as ( x ) approaches 0 exists and is finite, ( f(x) ) is differentiable at ( x = 0 ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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