Using the limit definition, how do you find the derivative of #f(x) =sqrt (x+1) #?

Answer 1

To find the derivative, apply the definition:

#f'(x) = lim_(h->0) (f(x+h) - f(x))/h#

With #f(x) = sqrt(x+1)#, we apply the definition:
#f'(x) = lim_(h->0) (sqrt(x+h+1) - sqrt(x+1))/h#

To handle the terms in the numerator, it looks like we will need to multiply by the conjugate:

#f'(x) =# #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))#
#f'(x) = lim_(h->0) ((x+h+1) - (x+1))/(h(sqrt(x+h+1)+sqrt(x+1))#
#f'(x) = lim_(h->0) cancel(h)/(cancel(h)(sqrt(x+h+1) + sqrt(x+1)))#
#f'(x) = lim_(h->0) 1/(sqrt(x+h+1) + sqrt(x+1))#

At this point, we can directly apply the limit to arrive at an answer:

#f'(x) = 1/(sqrt(x + 0 + 1) + sqrt(x + 1))#
#f'(x) = 1/(sqrt(x+1) + sqrt(x+1))#
#f'(x) = 1/(2sqrt(x+1))#

If you are familiar with the chain rule for derivatives, we can use it to test our result:

#f(x) = sqrt(x+1) = (x+1)^(1/2)# #f'(x) = (1/2)(x+1)^(1/2 - 1)# #f'(x) = (1/2)(x+1)^(-1/2)# #f'(x) = 1/(2sqrt(x+1))#
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Answer 2

To find the derivative of ( f(x) = \sqrt{x+1} ) using the limit definition, we use the formula:

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

First, we substitute ( f(x+h) = \sqrt{x+h+1} ) and ( f(x) = \sqrt{x+1} ) into the formula.

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

Next, we rationalize the numerator by multiplying the numerator and denominator by the conjugate of the numerator.

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

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

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

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

Now, we can take the limit as ( h \to 0 ):

[ f'(x) = \frac{1}{\sqrt{x+1} + \sqrt{x+1}} ]

[ f'(x) = \frac{1}{2\sqrt{x+1}} ]

So, the derivative of ( f(x) = \sqrt{x+1} ) with respect to ( x ) is ( f'(x) = \frac{1}{2\sqrt{x+1}} ).

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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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