# How do you find the derivative using quotient rule of #[x(3x+5)] / (1-x^2)#?

#f'(x)=(5x^2+6x+5)/(1-x^2)^2#

First, simplify the numerator.

Now, according to the quotient rule,

Find each derivative through the power rule.

Distribute and simplify.

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To find the derivative of ( \frac{x(3x+5)}{1-x^2} ) using the quotient rule, you follow this formula:

[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} ]

where ( u = x(3x+5) ) and ( v = (1-x^2) ).

Now, find the derivatives of ( u ) and ( v ):

( u' = (3x+5) + x(6) )

( v' = (-2x) )

Finally, plug these values into the quotient rule formula:

[ \frac{(1-x^2)(3x+5) - x(3x+5)(-2x)}{(1-x^2)^2} ]

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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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