# How do you find the derivative of # y=sqrt(x−3)# using the limit definition?

Then:

Using the limit definition:

Substitute in the functions:

Please observe that we have not changed the value of anything, because we have multiplied by 1 in a very special form.

Combine like terms in the numerator:

The h in the numerator cancels with the h in the denominator:

Now we can allow h to become 0:

Combine like terms:

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To find the derivative of ( y = \sqrt{x - 3} ) using the limit definition, we apply the definition of the derivative:

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

Substitute ( f(x) = \sqrt{x - 3} ) into the definition and simplify the expression:

[ f'(x) = \lim_{h \to 0} \frac{\sqrt{x + h - 3} - \sqrt{x - 3}}{h} ]

To eliminate the square roots in the numerator, we rationalize the expression by multiplying the numerator and denominator by the conjugate of the numerator:

[ f'(x) = \lim_{h \to 0} \frac{\sqrt{x + h - 3} - \sqrt{x - 3}}{h} \cdot \frac{\sqrt{x + h - 3} + \sqrt{x - 3}}{\sqrt{x + h - 3} + \sqrt{x - 3}} ]

[ f'(x) = \lim_{h \to 0} \frac{(x + h - 3) - (x - 3)}{h(\sqrt{x + h - 3} + \sqrt{x - 3})} ]

[ f'(x) = \lim_{h \to 0} \frac{x + h - 3 - x + 3}{h(\sqrt{x + h - 3} + \sqrt{x - 3})} ]

[ f'(x) = \lim_{h \to 0} \frac{h}{h(\sqrt{x + h - 3} + \sqrt{x - 3})} ]

[ f'(x) = \lim_{h \to 0} \frac{1}{\sqrt{x + h - 3} + \sqrt{x - 3}} ]

Now, as ( h ) approaches 0, we evaluate the limit:

[ f'(x) = \frac{1}{2\sqrt{x - 3}} ]

So, the derivative of ( y = \sqrt{x - 3} ) using the limit definition is ( \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.

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