# Using the limit definition, how do you differentiate #f(x) =4/(sqrt(x))#?

The limit definition of the derivative:

Thus, the derivative can be found through

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To differentiate ( f(x) = \frac{4}{\sqrt{x}} ) using the limit definition, we apply the definition of the derivative:

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

Substituting ( f(x) = \frac{4}{\sqrt{x}} ) into the formula, we have:

[ f'(x) = \lim_{h \to 0} \frac{\frac{4}{\sqrt{x + h}} - \frac{4}{\sqrt{x}}}{h} ]

To simplify, rationalize the numerator:

[ f'(x) = \lim_{h \to 0} \frac{4\sqrt{x} - 4\sqrt{x + h}}{h\sqrt{x}\sqrt{x + h}} ]

Combine the terms in the numerator:

[ f'(x) = \lim_{h \to 0} \frac{4(\sqrt{x} - \sqrt{x + h})}{h\sqrt{x}\sqrt{x + h}} ]

Now, to evaluate the limit, let's factor out ( \sqrt{x} ) from the numerator:

[ f'(x) = \lim_{h \to 0} \frac{4(\sqrt{x} - \sqrt{x + h})}{h\sqrt{x}\sqrt{x + h}} \cdot \frac{\sqrt{x} + \sqrt{x + h}}{\sqrt{x} + \sqrt{x + h}} ]

This simplifies to:

[ f'(x) = \lim_{h \to 0} \frac{4(x - (x + h))}{h\sqrt{x}(\sqrt{x} + \sqrt{x + h})} ]

[ f'(x) = \lim_{h \to 0} \frac{-4h}{h\sqrt{x}(\sqrt{x} + \sqrt{x + h})} ]

[ f'(x) = \lim_{h \to 0} \frac{-4}{\sqrt{x}(\sqrt{x} + \sqrt{x + h})} ]

[ f'(x) = \frac{-4}{2\sqrt{x} \cdot \sqrt{x}} ]

[ f'(x) = \frac{-2}{x} ]

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