# How do you differentiate #y = x * sqrt (4 - x^2)#?

# dy/dx =( 2 (2 - x^2 ))/sqrt(4 - x^2 ) #

Applying the "product rule" and "chain rule" to differentiate:

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To differentiate the function ( y = x \cdot \sqrt{4 - x^2} ), you can use the product rule and the chain rule.

First, let's denote ( u = x ) and ( v = \sqrt{4 - x^2} ).

Then, apply the product rule:

[ \frac{dy}{dx} = u'v + uv' ]

Where ( u' ) is the derivative of ( u ) with respect to ( x ) and ( v' ) is the derivative of ( v ) with respect to ( x ).

Now, calculate the derivatives:

[ u' = 1 ] [ v' = \frac{d}{dx}\left(\sqrt{4 - x^2}\right) = \frac{1}{2\sqrt{4 - x^2}} \cdot \frac{d}{dx}(4 - x^2) = \frac{-2x}{2\sqrt{4 - x^2}} = \frac{-x}{\sqrt{4 - x^2}} ]

Now, apply the product rule formula:

[ \frac{dy}{dx} = x \cdot \frac{-x}{\sqrt{4 - x^2}} + \sqrt{4 - x^2} \cdot 1 ]

[ \frac{dy}{dx} = \frac{-x^2}{\sqrt{4 - x^2}} + \sqrt{4 - x^2} ]

[ \frac{dy}{dx} = \frac{-x^2 + 4 - x^2}{\sqrt{4 - x^2}} ]

[ \frac{dy}{dx} = \frac{4 - 2x^2}{\sqrt{4 - x^2}} ]

So, the derivative of ( y = x \cdot \sqrt{4 - x^2} ) with respect to ( x ) is ( \frac{4 - 2x^2}{\sqrt{4 - x^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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