# What is the derivative of #2x(4-x^2)^(1/2)#?

Se calculations below.

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To find the derivative of ( 2x(4-x^2)^{\frac{1}{2}} ), you can use the product rule and the chain rule.

Let ( u = 2x ) and ( v = (4-x^2)^{\frac{1}{2}} ).

Applying the product rule ( (uv)' = u'v + uv' ) and the chain rule ( (g(h(x)))' = g'(h(x)) \cdot h'(x) ), the derivative is:

[ (2x)'(4-x^2)^{\frac{1}{2}} + 2x\left(\frac{1}{2}(4-x^2)^{-\frac{1}{2}}\cdot(-2x)\right) ]

[ = 2(4-x^2)^{\frac{1}{2}} + 2x\left(\frac{-x}{(4-x^2)^{\frac{1}{2}}}\right) ]

[ = 2(4-x^2)^{\frac{1}{2}} - \frac{2x^2}{(4-x^2)^{\frac{1}{2}}} ]

[ = \frac{2(4-x^2) - 2x^2}{(4-x^2)^{\frac{1}{2}}} ]

[ = \frac{8 - 4x^2}{(4-x^2)^{\frac{1}{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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