Using the limit definition, how do you differentiate #f(x)=(x+1)^(1/2)#?

Answer 1

#d/dx ( sqrt(x+1) ) = 1/(2sqrt(x+1))#

Based on the definition of derivative:

#(df)/dx = lim_(h->0) (f(x+h)-f(x))/h#
For #f(x) = (x+1)^(1/2) = sqrt(x+1)# we have then:
#d/dx ( sqrt(x+1) ) = lim_(h->0) ( sqrt(x+h+1)-sqrt(x+1))/h#

Now multiply numerator and denominator by the quantity:

#sqrt(x+h+1) + sqrt(x+1)#

and use the algebraic identity:

#(a-b)(a+b) = a^2-b^2#:
#d/dx ( sqrt(x+1) ) = lim_(h->0) (( sqrt(x+h+1)-sqrt(x+1))/h )((sqrt(x+h+1) + sqrt(x+1))/(sqrt(x+h+1) + sqrt(x+1)))#
#d/dx ( sqrt(x+1) ) = lim_(h->0) ( (x+h+1)-(x+1))/(h(sqrt(x+h+1) + sqrt(x+1))#
#d/dx ( sqrt(x+1) ) = lim_(h->0) cancel(h)/(cancel(h)(sqrt(x+h+1) + sqrt(x+1))#
#d/dx ( sqrt(x+1) ) = lim_(h->0) 1/(sqrt(x+h+1) + sqrt(x+1))#
#d/dx ( sqrt(x+1) ) = 1/(2sqrt(x+1))#
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Answer 2

To differentiate ( f(x) = (x+1)^{\frac{1}{2}} ) using the limit definition of the derivative, follow these steps:

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

  2. 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} ]

  3. 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}}} ]

  4. 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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Answer 3

To differentiate the function (f(x) = (x + 1)^{\frac{1}{2}}) using the limit definition of the derivative, you can follow these steps:

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

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

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

  4. Expand and simplify the expression.

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

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Answer from HIX Tutor

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.

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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