How do you find f'(x) using the limit definition given #f(x)=x^(1/2)#?
see below
Here we have
Here that means:
by some more manipulation, this then becomes the same as you would expect from a straight power rule
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To find ( f'(x) ) using the limit definition, where ( f(x) = x^{1/2} ), follow these steps:

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

Substitute the given function ( f(x) = x^{1/2} ) into the formula: [ f'(x) = \lim_{h \to 0} \frac{(x + h)^{1/2}  x^{1/2}}{h} ]

Simplify the expression: [ f'(x) = \lim_{h \to 0} \frac{(x + h)^{1/2}  x^{1/2}}{h} ] [ = \lim_{h \to 0} \frac{\frac{1}{\sqrt{x + h}}  \frac{1}{\sqrt{x}}}{h} ]

Rationalize the numerator: [ f'(x) = \lim_{h \to 0} \frac{\sqrt{x}  \sqrt{x + h}}{h\sqrt{x}\sqrt{x + h}} ]

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

Apply the conjugate rule: [ f'(x) = \lim_{h \to 0} \frac{(\sqrt{x}  \sqrt{x + h})(\sqrt{x} + \sqrt{x + h})}{h(\sqrt{x}\sqrt{x + h})(\sqrt{x} + \sqrt{x + h})} ]

Simplify the numerator: [ f'(x) = \lim_{h \to 0} \frac{x  (x + h)}{h(\sqrt{x}\sqrt{x + h})(\sqrt{x} + \sqrt{x + h})} ]

Cancel out ( h ): [ f'(x) = \lim_{h \to 0} \frac{h}{h(\sqrt{x}\sqrt{x + h})(\sqrt{x} + \sqrt{x + h})} ]

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

Evaluate the limit as ( h ) approaches ( 0 ): [ f'(x) = \frac{1}{2x^{3/2}} ]
Therefore, the derivative of ( f(x) = x^{1/2} ) is ( f'(x) = \frac{1}{2x^{3/2}} ).
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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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