# How do you find the derivative for #f(x) = (3 + 4x) /( 1 + x^2)#?

- Quotient rule states that
#(a/b)'=(a'b-ab')/b^2#

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To find the derivative of ( f(x) = \frac{3 + 4x}{1 + x^2} ), you can use the quotient rule. The quotient rule states that if you have a function ( f(x) = \frac{g(x)}{h(x)} ), then its derivative is given by:

[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} ]

For ( f(x) = \frac{3 + 4x}{1 + x^2} ):

[ f'(x) = \frac{(4)(1 + x^2) - (3 + 4x)(2x)}{(1 + x^2)^2} ]

[ f'(x) = \frac{4 + 4x^2 - 6x - 8x^2}{(1 + x^2)^2} ]

[ f'(x) = \frac{-4x^2 - 6x + 4}{(1 + x^2)^2} ]

So, the derivative of ( f(x) ) is ( f'(x) = \frac{-4x^2 - 6x + 4}{(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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