# How do you differentiate #y=(x^2+1)root3(x^2+2)#?

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To differentiate ( y = (x^2 + 1)\sqrt{3}(x^2 + 2) ), you can use the product rule. The product rule states that if ( y = u \cdot v ), where ( u ) and ( v ) are functions of ( x ), then ( \frac{dy}{dx} = u'v + uv' ).

Let ( u = x^2 + 1 ) and ( v = \sqrt{3}(x^2 + 2) ).

Now, find the derivatives of ( u ) and ( v ):

[ u' = 2x ] [ v' = \frac{d}{dx}[\sqrt{3}(x^2 + 2)] = \sqrt{3}(2x) ]

Apply the product rule:

[ \frac{dy}{dx} = (x^2 + 1) \cdot \sqrt{3}(2x) + 2x \cdot \sqrt{3}(x^2 + 2) ]

[ = 2x\sqrt{3}(x^2 + 1) + 2x\sqrt{3}(x^2 + 2) ]

[ = 2x\sqrt{3}(x^2 + 1 + x^2 + 2) ]

[ = 2x\sqrt{3}(2x^2 + 3) ]

So, ( \frac{dy}{dx} = 2x\sqrt{3}(2x^2 + 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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