# How do you find the derivative of #x^3(4x^2-1)^3#?

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To find the derivative of (x^3(4x^2-1)^3), you can use the chain rule. The derivative is (3x^2(4x^2-1)^3 + x^3 \cdot 3(4x^2-1)^2 \cdot 8x). This simplifies to (3x^2(4x^2-1)^2(4x^2-1+8x)).

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To find the derivative of ( x^3(4x^2-1)^3 ), you can use the product rule and the chain rule. The derivative of a product of functions ( u(x)v(x) ) is given by ( u'(x)v(x) + u(x)v'(x) ). The derivative of a function ( f(g(x)) ) where ( g(x) ) is another function is given by ( f'(g(x)) \cdot g'(x) ).

So, applying the product rule and chain rule to the given expression, we get:

[ \frac{d}{dx} [x^3(4x^2-1)^3] = 3x^2(4x^2-1)^3 + x^3 \cdot 3(4x^2-1)^2 \cdot 8x ]

Simplify the expression:

[ 3x^2(4x^2-1)^3 + 24x^4(4x^2-1)^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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