# How do you differentiate #f(x)=(1+arctanx)/(2-3arctanx)#?

Here we can use quotient rule.

Hence

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To differentiate ( f(x) = \frac{1 + \arctan(x)}{2 - 3\arctan(x)} ), you can use the quotient rule. The quotient rule states that if you have a function ( \frac{u(x)}{v(x)} ), then its derivative is given by ( \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} ).

In this case, ( u(x) = 1 + \arctan(x) ) and ( v(x) = 2 - 3\arctan(x) ).

Now, differentiate ( u(x) ) and ( v(x) ) separately.

( u'(x) = \frac{1}{1 + x^2} ) and ( v'(x) = \frac{-3}{1 + x^2} ).

Now, apply the quotient rule:

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

Simplify this expression to get the derivative of ( f(x) ).

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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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