What is the derivative of #x * e^3x + tan^1 2x#?
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To find the derivative of the given function, (x \cdot e^{3x} + \tan^{1}(2x)), we differentiate each term separately using the rules of differentiation.

For the term (x \cdot e^{3x}): Apply the product rule: [ \frac{d}{dx}\left(x \cdot e^{3x}\right) = x \cdot \frac{d}{dx}(e^{3x}) + \frac{d}{dx}(x) \cdot e^{3x} ] [ = x \cdot (3e^{3x}) + e^{3x} ] [ = 3xe^{3x} + e^{3x} ]

For the term (\tan^{1}(2x)): Apply the chain rule: [ \frac{d}{dx}(\tan^{1}(2x)) = \frac{1}{1 + (2x)^2} \cdot \frac{d}{dx}(2x) ] [ = \frac{2}{1 + 4x^2} ]
Therefore, the derivative of the function (x \cdot e^{3x} + \tan^{1}(2x)) is (3xe^{3x} + e^{3x} + \frac{2}{1 + 4x^2}).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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