How do you use the formal definition of differentiation as a limit to find the derivative of #f(x)=1/(x1)#?
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Firstly, let's remember the limit definition of the derivative :
That's it.
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To find the derivative of ( f(x) = \frac{1}{x1} ) using the formal definition of differentiation, follow these steps:

Start with the formal definition of the derivative: ( f'(x) = \lim_{h \to 0} \frac{f(x + h)  f(x)}{h} ).

Substitute ( f(x) = \frac{1}{x1} ) into the definition.

Simplify the expression using algebraic manipulation.

Take the limit as ( h ) approaches 0 to find the derivative.
Applying these steps will yield the derivative of ( f(x) = \frac{1}{x1} ) using the formal definition of differentiation.
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To use the formal definition of differentiation as a limit to find the derivative of ( f(x) = \frac{1}{x  1} ), we start with the definition of the derivative:
[ f'(x) = \lim_{h \to 0} \frac{f(x + h)  f(x)}{h} ]
Substitute ( f(x) = \frac{1}{x  1} ) into this definition:
[ f'(x) = \lim_{h \to 0} \frac{\frac{1}{x + h  1}  \frac{1}{x  1}}{h} ]
Combine the fractions:
[ f'(x) = \lim_{h \to 0} \frac{(x  1)  (x + h  1)}{h(x  1)(x + h  1)} ]
[ f'(x) = \lim_{h \to 0} \frac{x  1  x  h + 1}{h(x  1)(x + h  1)} ]
[ f'(x) = \lim_{h \to 0} \frac{h}{h(x  1)(x + h  1)} ]
[ f'(x) = \lim_{h \to 0} \frac{1}{(x  1)(x + h  1)} ]
Now, we can evaluate the limit as ( h ) approaches 0:
[ f'(x) = \frac{1}{(x  1)^2} ]
Therefore, the derivative of ( f(x) = \frac{1}{x  1} ) with respect to ( x ) is ( f'(x) = \frac{1}{(x  1)^2} ).
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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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