# What is the derivative of #y=ln[x/(2x+3)]^(1/2)#?

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To find the derivative of ( y = \ln\left[\frac{x}{2x+3}\right]^{1/2} ), use the chain rule and the power rule for differentiation. The derivative is:

[ \frac{dy}{dx} = \frac{1}{2}\left[\frac{1}{\frac{x}{2x+3}}\right] \cdot \frac{d}{dx}\left(\frac{x}{2x+3}\right) ]

[ = \frac{1}{2}\left[\frac{1}{\frac{x}{2x+3}}\right] \cdot \left(\frac{(2x+3) - x(2)}{(2x+3)^2}\right) ]

[ = \frac{1}{2}\left[\frac{1}{\frac{x}{2x+3}}\right] \cdot \left(\frac{3}{(2x+3)^2}\right) ]

[ = \frac{1}{2}\left(\frac{2x+3}{x}\right) \cdot \left(\frac{3}{(2x+3)^2}\right) ]

[ = \frac{3}{2x(2x+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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