# How do you differentiate #f(x) = (x-1)(x-2)# using the product rule?

The product rule states that for a function

Let's examine each of these functions separately.

According to the power rule:

and

We are now able to use the product rule.

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To differentiate ( f(x) = (x-1)(x-2) ) using the product rule:

- Identify the two functions being multiplied: ( f(x) = (x-1) ) and ( g(x) = (x-2) ).
- Apply the product rule formula: ( (f \cdot g)' = f' \cdot g + f \cdot g' ).
- Differentiate each function individually:
- ( f'(x) = 1 ) (differentiation of ( x-1 ))
- ( g'(x) = 1 ) (differentiation of ( x-2 )).

- Substitute these differentials into the product rule formula: ( (f \cdot g)' = (1)(x-2) + (x-1)(1) ).
- Simplify the expression: ( (f \cdot g)' = x - 2 + x - 1 ).
- Combine like terms: ( (f \cdot g)' = 2x - 3 ).

Therefore, the derivative of ( f(x) = (x-1)(x-2) ) using the product rule is ( f'(x) = 2x - 3 ).

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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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