How to use the alternate definition to find the derivative of #f(x)=sqrt(x+3)# at x=1?

Answer 1

Here are a couple ways you can do the limit calculation for the derivative. Both methods involve "rationalizing the numerator" (not the denominator) as a trick to help you calculate the limits.

#f'(1)=lim_{h->0}\frac{f(1+h)-f(1)}{h}#
#=lim_{h->0}\frac{\sqrt{4+h}-2}{h}\cdot \frac{\sqrt{4+h}+2}{\sqrt{4+h}+2}#
#=lim_{h->0}\frac{4+h-4}{h(\sqrt{4+h}+2)}=lim_{h->0}\frac{1}{\sqrt{4+h}+2}=\frac{1}{4}#

OR

#f'(1)=\lim_{x->1}\frac{f(x)-f(1)}{x-1}=\lim_{x->1}\frac{\sqrt{x+3}-2}{x-1}#
#=lim_{x->1}\frac{x+3-4}{(x-1)(\sqrt{x+3}+2)}#
#=lim_{x->1}\frac{1}{sqrt{x+3}+2}=\frac{1}{4}#
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Answer 2

To find the derivative of ( f(x) = \sqrt{x + 3} ) at ( x = 1 ) using the alternate definition, follow these steps:

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

  2. Substitute ( x = 1 ) into the function: [ f(1) = \sqrt{1 + 3} = \sqrt{4} = 2 ]

  3. Substitute ( x = 1 ) and ( h = 0 ) into the alternate definition: [ f'(1) = \lim_{h \to 0} \frac{f(1 + h) - f(1)}{h} ]

  4. Substitute ( f(1) = 2 ) into the expression: [ f'(1) = \lim_{h \to 0} \frac{\sqrt{1 + h + 3} - 2}{h} ]

  5. Simplify the expression: [ f'(1) = \lim_{h \to 0} \frac{\sqrt{h + 4} - 2}{h} ]

  6. Rationalize the numerator: [ f'(1) = \lim_{h \to 0} \frac{\sqrt{h + 4} - 2}{h} \times \frac{\sqrt{h + 4} + 2}{\sqrt{h + 4} + 2} ] [ f'(1) = \lim_{h \to 0} \frac{h}{h(\sqrt{h + 4} + 2)} ]

  7. Cancel out ( h ): [ f'(1) = \lim_{h \to 0} \frac{1}{\sqrt{h + 4} + 2} ]

  8. Evaluate the limit as ( h ) approaches ( 0 ): [ f'(1) = \frac{1}{\sqrt{4} + 2} = \frac{1}{2 + 2} = \frac{1}{4} ]

Therefore, the derivative of ( f(x) = \sqrt{x + 3} ) at ( x = 1 ) using the alternate definition is ( \frac{1}{4} ).

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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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