# How do you use the definition of a derivative to find the derivative of #f(x) = 7/(9x)#?

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To find the derivative of ( f(x) = \frac{7}{9x} ) using the definition of a derivative, we first write down the definition:

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

Then we substitute ( f(x) = \frac{7}{9x} ) into the definition:

[ f'(x) = \lim_{h \to 0} \frac{\frac{7}{9(x + h)} - \frac{7}{9x}}{h} ]

Next, we simplify the expression:

[ f'(x) = \lim_{h \to 0} \frac{\frac{7}{9(x + h)} - \frac{7}{9x}}{h} = \lim_{h \to 0} \frac{7x - 7(x + h)}{9x(x + h)h} ]

[ = \lim_{h \to 0} \frac{7x - 7x - 7h}{9x(x + h)h} ]

[ = \lim_{h \to 0} \frac{-7h}{9x(x + h)h} ]

[ = \lim_{h \to 0} \frac{-7}{9x(x + h)} ]

Finally, we take the limit as ( h ) approaches 0:

[ f'(x) = \frac{-7}{9x^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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