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

The answer is

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To differentiate ( f(x) = x^2 \sqrt{x} ) using the product rule:

- Identify the two functions being multiplied: ( f(x) = u(x) \cdot v(x) ), where ( u(x) = x^2 ) and ( v(x) = \sqrt{x} ).
- Apply the product rule: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
- Find the derivatives of each function:
- ( u'(x) = 2x ) (derivative of ( x^2 )).
- ( v'(x) = \frac{1}{2\sqrt{x}} ) (derivative of ( \sqrt{x} )).

- Substitute the derivatives and original functions into the product rule formula: ( f'(x) = (2x) \cdot \sqrt{x} + (x^2) \cdot \frac{1}{2\sqrt{x}} ).
- Simplify the expression: ( f'(x) = 2x\sqrt{x} + \frac{x^2}{2\sqrt{x}} ).
- If needed, further simplify the expression by rationalizing the denominator: ( f'(x) = 2x\sqrt{x} + \frac{x^2\sqrt{x}}{2x} ). ( f'(x) = 2x\sqrt{x} + \frac{x\sqrt{x}}{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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