# How do you differentiate #f(x)=(x+1)/sqrtx# using the quotient rule?

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To differentiate ( f(x) = \frac{x+1}{\sqrt{x}} ) using the quotient rule:

- Identify the numerator function: ( u(x) = x + 1 )
- Identify the denominator function: ( v(x) = \sqrt{x} )
- Apply the quotient rule formula: [ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} ]
- Calculate the derivatives: [ u'(x) = 1 ] [ v'(x) = \frac{1}{2\sqrt{x}} ]
- Plug the derivatives and functions into the quotient rule formula: [ f'(x) = \frac{(1)(\sqrt{x}) - (x+1)\left(\frac{1}{2\sqrt{x}}\right)}{(\sqrt{x})^2} ]
- Simplify the expression: [ f'(x) = \frac{\sqrt{x} - \frac{x+1}{2\sqrt{x}}}{x} ] [ f'(x) = \frac{2x\sqrt{x} - (x+1)}{2x\sqrt{x}} ] [ f'(x) = \frac{2x\sqrt{x} - x - 1}{2x\sqrt{x}} ]

So, the derivative of ( f(x) = \frac{x+1}{\sqrt{x}} ) using the quotient rule is ( f'(x) = \frac{2x\sqrt{x} - x - 1}{2x\sqrt{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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