Using the limit definition, how do you differentiate #f(x)=(x+1)^(1/2)#?
Based on the definition of derivative:
Now multiply numerator and denominator by the quantity:
and use the algebraic identity:
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To differentiate ( f(x) = (x+1)^{\frac{1}{2}} ) using the limit definition of the derivative, follow these steps:

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

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

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

Simplify further: [ f'(x) = \lim_{h \to 0} \frac{x + h + 1  x  1}{h((x+h)+1)^{\frac{1}{2}} + h(x+1)^{\frac{1}{2}}} ] [ = \lim_{h \to 0} \frac{h}{h((x+h)+1)^{\frac{1}{2}} + h(x+1)^{\frac{1}{2}}} ] [ = \lim_{h \to 0} \frac{1}{((x+h)+1)^{\frac{1}{2}} + (x+1)^{\frac{1}{2}}} ] [ = \frac{1}{(x+1)^{\frac{1}{2}} + (x+1)^{\frac{1}{2}}} ] [ = \frac{1}{2(x+1)^{\frac{1}{2}}} ]
Therefore, the derivative of ( f(x) = (x+1)^{\frac{1}{2}} ) using the limit definition is ( f'(x) = \frac{1}{2(x+1)^{\frac{1}{2}}} ).
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To differentiate the function (f(x) = (x + 1)^{\frac{1}{2}}) using the limit definition of the derivative, you can follow these steps:

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

Substitute the function (f(x) = (x + 1)^{\frac{1}{2}}) into the formula.

Simplify the expression (f(x + h)  f(x)) using the given function.

Expand and simplify the expression.

Take the limit as (h) approaches 0 to find the derivative (f'(x)).
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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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