# How do you use the definition of a derivative to find the derivative of #f(x)=(1/x)-2#?

I found

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To find the derivative of ( f(x) = \frac{1}{x} - 2 ), we use the definition of a derivative, which states:

[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

By substituting the given function ( f(x) ), we have:

[ f'(x) = \lim_{h \to 0} \frac{\left(\frac{1}{x + h} - 2\right) - \left(\frac{1}{x} - 2\right)}{h} ]

After simplifying, we get:

[ f'(x) = \lim_{h \to 0} \frac{\frac{1}{x + h} - \frac{1}{x}}{h} ]

Using the properties of fractions, we find a common denominator:

[ f'(x) = \lim_{h \to 0} \frac{x - (x + h)}{x(x + h)h} ]

Simplify the numerator:

[ f'(x) = \lim_{h \to 0} \frac{-h}{x(x + h)h} ]

[ f'(x) = \lim_{h \to 0} \frac{-1}{x(x + h)} ]

Now, as ( h ) approaches 0, the expression converges to:

[ f'(x) = \frac{-1}{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.

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