# Using the limit definition, how do you find the derivative of #f(x)=sqrt(2x-1)#?

If the limit exists, this definition is called the First Principles of Derivatives .

#lim_{h->0} frac{sqrt{2x+2h-1}-sqrt{2x-1}}{h} = lim_{h->0} frac{ ( sqrt{2x+2h-1}-sqrt{2x-1} )( sqrt{2x+2h-1}+sqrt{2x-1} ) }{ h(sqrt{2x+2h-1}+sqrt{2x-1}) }#

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To find the derivative of ( f(x) = \sqrt{2x - 1} ) using the limit definition, you first express ( f(x) ) as ( \frac{f(x + h) - f(x)}{h} ). Then, simplify and take the limit of this expression as ( h ) approaches 0. The derivative is the result of this limit calculation.

- Start with ( f(x) = \sqrt{2x - 1} ).
- Determine ( f(x + h) ) by substituting ( x + h ) into ( f(x) ). ( f(x + h) = \sqrt{2(x + h) - 1} ).
- Find ( \frac{f(x + h) - f(x)}{h} ). ( \frac{f(x + h) - f(x)}{h} = \frac{\sqrt{2(x + h) - 1} - \sqrt{2x - 1}}{h} ).
- Rationalize the numerator by multiplying by the conjugate.
- Simplify the expression and then take the limit of this expression as ( h ) approaches 0.

After simplification and taking the limit, you'll obtain the derivative of ( f(x) ).

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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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