Using the limit definition, how do you find the derivative of #f(x) =sqrt (x+1) #?
To find the derivative, apply the definition:
#f'(x) = lim_(h->0) (f(x+h) - f(x))/h#
To handle the terms in the numerator, it looks like we will need to multiply by the conjugate:
At this point, we can directly apply the limit to arrive at an answer:
If you are familiar with the chain rule for derivatives, we can use it to test our result:
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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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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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