# How do I use the limit definition of derivative to find #f'(x)# for #f(x)=1/sqrt(x)# ?

Find the common denominator

Consolidate the numerator of the complex fraction.

Dividing fractions is equivalent to multiplying by the reciprocal

Rationalize the numerator

Simplify. Remember difference of perfect squares.

Distribute the negative in the numerator

Manipulate the exponents

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To use the limit definition of the derivative to find ( f'(x) ) for ( f(x) = \frac{1}{\sqrt{x}} ), 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}{\sqrt{x}} ) into the definition:

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

To simplify, rationalize the numerator:

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

Now, factor out ( \sqrt{x} ) from the numerator:

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

Apply the conjugate rule to simplify the numerator:

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

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

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

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

Now, as ( h ) approaches 0, the denominator remains finite, so we can evaluate the limit:

[ f'(x) = \frac{-1}{2x^{3/2}} ]

Therefore, the derivative of ( f(x) = \frac{1}{\sqrt{x}} ) is ( f'(x) = \frac{-1}{2x^{3/2}} ).

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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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