# How do you find the derivative of #f(x) = 1/sqrt(2x-1)# by first principles?

derivative of function to a power

The factor 2 comes from the derivative of f itself

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Use limit definition of derivative to find:

#f'(x) = -(2x-1)^(-3/2)#

So:

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To find the derivative of ( f(x) = \frac{1}{\sqrt{2x - 1}} ) using first principles, we start with the definition of the derivative:

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

First, we need to find ( f(x + h) ):

[ f(x + h) = \frac{1}{\sqrt{2(x + h) - 1}} ]

Now, substitute ( f(x + h) ) and ( f(x) ) into the difference quotient:

[ f'(x) = \lim_{h \to 0} \frac{\frac{1}{\sqrt{2(x + h) - 1}} - \frac{1}{\sqrt{2x - 1}}}{h} ]

Next, we rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator:

[ f'(x) = \lim_{h \to 0} \frac{\sqrt{2x - 1} - \sqrt{2(x + h) - 1}}{h\sqrt{2(x + h) - 1}\sqrt{2x - 1}} ]

Now, combine the terms in the numerator:

[ f'(x) = \lim_{h \to 0} \frac{\sqrt{2x - 1} - \sqrt{2x + 2h - 1}}{h\sqrt{2(x + h) - 1}\sqrt{2x - 1}} ]

To simplify further, factor out ( -1 ) from the square root:

[ f'(x) = \lim_{h \to 0} \frac{\sqrt{2x - 1} - \sqrt{2x - 1 - 2h}}{h\sqrt{2(x + h) - 1}\sqrt{2x - 1}} ]

Now, we can cancel out the ( h ) terms:

[ f'(x) = \lim_{h \to 0} \frac{\sqrt{2x - 1} - \sqrt{2x - 1 - 2h}}{h} \cdot \frac{1}{\sqrt{2(x + h) - 1}\sqrt{2x - 1}} ]

Finally, take the limit as ( h ) approaches 0:

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

Therefore, the derivative of ( f(x) = \frac{1}{\sqrt{2x - 1}} ) by first principles is ( f'(x) = \frac{-1}{2(x - 1)^{3/2}} ).

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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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