# How do you find the derivative of #y=2/(3x^2)#?

Applying the power rule:

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To find the derivative of ( y = \frac{2}{3x^2} ), you can use the power rule for differentiation. The power rule states that if you have a function of the form ( f(x) = ax^n ), then its derivative is ( f'(x) = nax^{n-1} ). Applying this rule to the given function ( y = \frac{2}{3x^2} ), you get:

[ y' = \frac{d}{dx} \left( \frac{2}{3x^2} \right) ] [ = \frac{2}{3} \cdot \frac{d}{dx} \left( x^{-2} \right) ] [ = \frac{2}{3} \cdot (-2x^{-3}) ] [ = -\frac{4}{3x^3} ]

Therefore, the derivative of ( y = \frac{2}{3x^2} ) is ( y' = -\frac{4}{3x^3} ).

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To find the derivative of ( y = \frac{2}{3x^2} ), you can use the power rule for differentiation, which states that if ( f(x) = x^n ), then ( f'(x) = nx^{n-1} ).

- Rewrite the function as ( y = 2 \cdot (3x^2)^{-1} ).
- Apply the power rule to differentiate ( 3x^2 ) with respect to ( x ): ( \frac{d}{dx}(3x^2) = 2 \cdot 3 \cdot x^{2-1} = 6x ).
- Apply the chain rule to differentiate ( (3x^2)^{-1} ): ( \frac{d}{dx}\left((3x^2)^{-1}\right) = -1 \cdot (3x^2)^{-2} \cdot 6x ).
- Simplify the expression: ( \frac{d}{dx}\left((3x^2)^{-1}\right) = -\frac{6x}{(3x^2)^2} = -\frac{6x}{9x^4} = -\frac{2x}{3x^4} ).
- Multiply the derivative by the constant factor ( 2 ): ( \frac{d}{dx}\left(\frac{2}{3x^2}\right) = 2 \cdot -\frac{2x}{3x^4} = -\frac{4x}{3x^4} ).

Therefore, the derivative of ( y = \frac{2}{3x^2} ) with respect to ( x ) is ( -\frac{4x}{3x^4} ).

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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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