# How do you find the derivative for #y= (2x) / (sqrt(x - 1))#?

You can use two approaches to differentiate this function.

This means that you can approach it in two different ways, either by using the product rule or by using the quotient rule.

The chain rule will be needed in both cases, though.

To keep the answer from becoming too long, I will only show you one of the two approaches.

So, your function can be written as

The quotient rule allows you to differentiate this function by using

This means that you have

This will get you

Your target derivative will now be

You can simplify this by using fractional exponents

The starting point for the product rule approach would have been

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To find the derivative of ( y = \frac{2x}{\sqrt{x - 1}} ), you can use the quotient rule, which states that if ( y = \frac{u}{v} ), then ( y' = \frac{u'v - uv'}{v^2} ).

Let ( u = 2x ) and ( v = \sqrt{x - 1} ). Then, ( u' = 2 ) and ( v' = \frac{1}{2\sqrt{x - 1}} ).

Now, applying the quotient rule:

[ y' = \frac{(2)(\sqrt{x - 1}) - (2x)\left(\frac{1}{2\sqrt{x - 1}}\right)}{(\sqrt{x - 1})^2} ]

Simplify the expression:

[ y' = \frac{2\sqrt{x - 1} - \frac{2x}{2\sqrt{x - 1}}}{x - 1} ]

[ y' = \frac{2\sqrt{x - 1} - \frac{x}{\sqrt{x - 1}}}{x - 1} ]

[ y' = \frac{2\sqrt{x - 1} - \frac{x}{\sqrt{x - 1}}}{x - 1} ]

[ y' = \frac{2\sqrt{x - 1} - \frac{x}{\sqrt{x - 1}}}{x - 1} ]

[ y' = \frac{2\sqrt{x - 1} - \frac{x}{\sqrt{x - 1}}}{x - 1} ]

[ y' = \frac{2\sqrt{x - 1} - \frac{x}{\sqrt{x - 1}}}{x - 1} ]

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

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

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

So, the derivative of ( y = \frac{2x}{\sqrt{x - 1}} ) is ( y' = \frac{x - 2}{(x - 1)^{\frac{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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