How do you differentiate #f(x)=arctanx*3x# using the product rule?
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To differentiate the function (f(x) = \arctan(x) \cdot 3x) using the product rule, we apply the formula: ((uv)' = u'v + uv'), where (u = \arctan(x)) and (v = 3x).
First, we find the derivatives of (u) and (v):
(u' = \frac{1}{1 + x^2}) (derivative of (\arctan(x)) with respect to (x))
(v' = 3) (derivative of (3x) with respect to (x))
Then, we substitute these derivatives into the product rule formula:
(f'(x) = u'v + uv' = \frac{1}{1 + x^2} \cdot 3x + \arctan(x) \cdot 3)
Thus, the derivative of (f(x) = \arctan(x) \cdot 3x) with respect to (x) using the product rule is (f'(x) = \frac{3x}{1 + x^2} + 3\arctan(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.
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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