How do you use the definition of a derivative to find the derivative of #f(x) = sqrtx + 2# to calculate f'(2)?

Answer 1

#[d/dx (sqrt(x) + 2)]_(x=2) = 1/(2sqrt2)#

By definition of derivative:

#f'(2) = lim_(h->0) (f(2+h)-f(2))/h#
For #f(x) = sqrt(x) + 2# this is:
#f'(2) = lim_(h->0) (sqrt(2+h) + 2 -sqrt(2) -2)/h#

simplify:

#f'(2)= lim_(h->0) (sqrt(2+h) -sqrt(2) )/h#
rationalize the numerator using the identity #(a+b)(a-b) = (a^2-b^2)#:
#f'(2)= lim_(h->0) ((sqrt(2+h) -sqrt(2) )/h )( (sqrt(2+h) + sqrt(2) )/(sqrt(2+h) +sqrt(2) ))#
#f'(2) = lim_(h->0) (2+h -2) /(h (sqrt(2+h) +sqrt(2) ))#
#f'(2) = lim_(h->0) h /(h (sqrt(2+h) +sqrt(2) ))#
#f'(2) = lim_(h->0) 1 / (sqrt(2+h) +sqrt(2) )#
#f'(2) = 1/(2sqrt(2))#
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Answer 2

To find the derivative of ( f(x) = \sqrt{x} + 2 ), we use the definition of a derivative, which states:

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

Plugging in ( f(x) = \sqrt{x} + 2 ), we have:

[ f'(x) = \lim_{h \to 0} \frac{\sqrt{x+h} + 2 - (\sqrt{x} + 2)}{h} ]

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

To simplify this expression, we use the conjugate pair:

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

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

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

Now, we plug in ( x = 2 ) to find ( f'(2) ):

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

Taking the limit as ( h \to 0 ):

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

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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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