# What is the first derivative and second derivative of #4x^(1/3)+2x^(4/3)#?

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The first derivative is ( \frac{4}{3}x^{-\frac{2}{3}} + \frac{8}{3}x^{\frac{1}{3}} ), and the second derivative is ( -\frac{8}{9}x^{-\frac{5}{3}} + \frac{8}{9}x^{-\frac{2}{3}} ).

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To find the first derivative of (4x^{1/3} + 2x^{4/3}), you differentiate each term separately using the power rule for derivatives. The power rule states that if you have a term of the form (ax^n), the derivative is (anx^{n-1}). Applying this rule:

For (4x^{1/3}), the derivative is ( \frac{d}{dx}(4x^{1/3}) = 4 \cdot \frac{1}{3} x^{(1/3)-1} = \frac{4}{3}x^{-2/3}).

For (2x^{4/3}), the derivative is ( \frac{d}{dx}(2x^{4/3}) = 2 \cdot \frac{4}{3} x^{(4/3)-1} = \frac{8}{3}x^{1/3}).

Therefore, the first derivative of (4x^{1/3} + 2x^{4/3}) is (\frac{4}{3}x^{-2/3} + \frac{8}{3}x^{1/3}).

To find the second derivative, you differentiate the first derivative with respect to (x) again:

For (\frac{4}{3}x^{-2/3}), the second derivative is ( \frac{d}{dx}\left(\frac{4}{3}x^{-2/3}\right) = \frac{4}{3} \cdot \left(-\frac{2}{3}\right)x^{(-2/3)-1} = -\frac{8}{9}x^{-5/3}).

For (\frac{8}{3}x^{1/3}), the second derivative is ( \frac{d}{dx}\left(\frac{8}{3}x^{1/3}\right) = \frac{8}{3} \cdot \frac{1}{3}x^{(1/3)-1} = \frac{8}{9}x^{-2/3}).

Therefore, the second derivative of (4x^{1/3} + 2x^{4/3}) is (-\frac{8}{9}x^{-5/3} + \frac{8}{9}x^{-2/3}).

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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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