Using the limit definition, how do you differentiate #f(x) =sqrt(x−3)#?
See the explanation section below.
The crucial step is to use the following to remove the square roots from the numerator.
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Using the limit definition of the derivative, the derivative of f(x) = sqrt(x - 3) is:
f'(x) = lim[h->0] [(sqrt(x + h - 3) - sqrt(x - 3)) / h]
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To differentiate ( f(x) = \sqrt{x - 3} ) using the limit definition of the derivative, we start by finding the difference quotient:
[ \frac{f(x + h) - f(x)}{h} ]
Substitute ( f(x) = \sqrt{x - 3} ) into the difference quotient:
[ \frac{\sqrt{x + h - 3} - \sqrt{x - 3}}{h} ]
To simplify this expression, we multiply both the numerator and denominator by the conjugate of the numerator:
[ \frac{(\sqrt{x + h - 3} - \sqrt{x - 3})(\sqrt{x + h - 3} + \sqrt{x - 3})}{h(\sqrt{x + h - 3} + \sqrt{x - 3})} ]
Expanding the numerator and canceling out common terms yields:
[ \frac{(x + h - 3) - (x - 3)}{h(\sqrt{x + h - 3} + \sqrt{x - 3})} ]
Simplifying further gives:
[ \frac{h}{h(\sqrt{x + h - 3} + \sqrt{x - 3})} ]
Now, as ( h ) approaches 0, the expression becomes:
[ \lim_{h \to 0} \frac{1}{\sqrt{x + h - 3} + \sqrt{x - 3}} ]
Now, evaluate the limit as ( h ) approaches 0:
[ \lim_{h \to 0} \frac{1}{\sqrt{x + h - 3} + \sqrt{x - 3}} = \frac{1}{2\sqrt{x - 3}} ]
Therefore, the derivative of ( f(x) = \sqrt{x - 3} ) using the limit definition is:
[ f'(x) = \frac{1}{2\sqrt{x - 3}} ]
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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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- What is the equation of the normal line of #f(x)=sqrt(2x^2-x)# at #x=-1#?
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