# What is the derivative of #f(x) = x(2-x)^2#?

Konstantios's answer provides some important concepts but for this particular problem there may be a simpler method.

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The derivative using the product rule is

#(df)/dx=x'*(2-x)^2+x*[(2-x)^2]'=(2-x)^2+2*x*(-1)*(2-x)=(2-x)^2-2x*(2-x)=
=3x^2-8x+4#

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To find the derivative of the function ( f(x) = x(2-x)^2 ), you can use the product rule and the chain rule. Applying these rules:

[ f'(x) = (1)(2-x)^2 + x \cdot 2(2-x)(-1) ] [ f'(x) = (2-x)^2 - 2x(2-x) ] [ f'(x) = (2-x)^2 - 4x(2-x) ] [ f'(x) = (2-x)^2 - 4x(2) + 4x^2 ] [ f'(x) = (2-x)^2 - 8x + 4x^2 ] [ f'(x) = 4x^2 - 4x + 4 - 8x + 4x^2 ] [ f'(x) = 8x^2 - 12x + 4 ]

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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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