# How do you differentiate #((9y^2)(-2y^2))/(7y^4)#?

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To differentiate the expression (\frac{{9y^2 \cdot (-2y^2)}}{{7y^4}}), apply the quotient rule for differentiation. The quotient rule states that if you have a quotient of two functions (f(x)) and (g(x)), then the derivative of the quotient is given by (\frac{{f'(x)g(x) - f(x)g'(x)}}{{[g(x)]^2}}).

Let (f(y) = 9y^2 \cdot (-2y^2)) and (g(y) = 7y^4).

First, find the derivatives of (f(y)) and (g(y)). (f'(y) = (18y)(-2y^2) + (9y^2)(-4y) = -36y^3 - 36y^3 = -72y^3) (g'(y) = 28y^3)

Now apply the quotient rule: (\frac{{f'(y)g(y) - f(y)g'(y)}}{{[g(y)]^2}} = \frac{{-72y^3 \cdot 7y^4 - (9y^2 \cdot (-2y^2)) \cdot 28y^3}}{{(7y^4)^2}})

Simplify: (\frac{{-504y^7 - (-18y^4 \cdot 28y^3)}}{{49y^8}}) (\frac{{-504y^7 + 504y^7}}{{49y^8}} = \frac{{0}}{{49y^8}})

So, the derivative of (\frac{{9y^2 \cdot (-2y^2)}}{{7y^4}}) is (0).

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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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