# What is the derivative of #xsqrt(1-x)#?

The derivative is

#d(x*sqrt(1-x))/dx=1*sqrt(1-x)+x*(1/2*1/(sqrt(1-x))*(-1))=
sqrt(1-x)-x/(2sqrt(1-x))#

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To find the derivative of (x\sqrt{1-x}), you can use the product rule. The product rule states that if you have two functions (u(x)) and (v(x)), then the derivative of their product (u(x)v(x)) is given by (u'(x)v(x) + u(x)v'(x)).

Let (u(x) = x) and (v(x) = \sqrt{1-x}). Then, the derivative of (u(x)) with respect to (x) is (u'(x) = 1) and the derivative of (v(x)) with respect to (x) is (v'(x) = -\frac{1}{2\sqrt{1-x}}).

Applying the product rule:

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

So, the derivative of (x\sqrt{1-x}) with respect to (x) is (\sqrt{1-x} - \frac{x}{2\sqrt{1-x}}).

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The derivative of ( x\sqrt{1-x} ) is ( \frac{d}{dx}(x\sqrt{1-x}) = \sqrt{1-x} - \frac{x}{2\sqrt{1-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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