# How do you differentiate #f(x)= (3x-2)^10 * (5x^2-x+1)^12# using the product rule?

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To differentiate the function ( f(x) = (3x - 2)^{10} \cdot (5x^2 - x + 1)^{12} ) using the product rule, you apply the formula:

[ (uv)' = u'v + uv' ]

Where ( u = (3x - 2)^{10} ) and ( v = (5x^2 - x + 1)^{12} ). Then, you find the derivatives ( u' ) and ( v' ) and substitute them into the formula:

[ u' = 10(3x - 2)^9 \cdot 3 ] [ v' = 12(5x^2 - x + 1)^{11} \cdot (10x - 1) ]

Substitute these derivatives into the product rule formula:

[ f'(x) = (10(3x - 2)^9 \cdot 3) \cdot (5x^2 - x + 1)^{12} + (3x - 2)^{10} \cdot (12(5x^2 - x + 1)^{11} \cdot (10x - 1)) ]

Simplify this expression to get the derivative of the function ( f(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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